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CAMBRIDGE    PHYSICAL    SERIES 

GENERAL  EDITORS  : — A.   HUTCHINSON,  M.A. 
AND  W.  C.  D.  WHETHAM,  M.A.,  F.R.S. 


MECHANICS 


CAMBRIDGE   UNIVERSITY  PRESS 

HonUon:    FETTER  LANE,   E.G. 

C.  F.  CLAY,  MANAGER 


CEbiniurgfj:  100,  PRINCES  STREET 

ALSO 

HenUon:  H.  K.  LEWIS,  136,  GOWER  STREET,  W.C. 

Berlin:  A.  ASHER  AND  CO. 

Eetpjig:    F.   A.   BROCKHAUS 

£tto  gork:   G.  P.  PUTNAM'S  SONS 

anb  Calcutta:  MACMILLAN  AND  CO.,  LTD. 


All  rights  reserved 


ARCHIMEDES 


MECHANICS 


BY 


JOHN  COX,  M.A.,   F.R.S.C. 

HONORARY   LL.D.,   QUEEN'S   UNIVERSITY,    KINGSTON  J 

FORMERLY   PROFESSOR   OF   PHYSICS   IN   MCGILL   UNIVERSITY,   MONTREAL; 
SOMETIME   FELLOW   OF   TRINITY   COLLEGE,    CAMBRIDGE. 


CAMBRIDGE: 

AT    THE    UNIVERSITY    PRESS. 
1909 


if 


First  Edition  1904 
Reprinted  1909 


TO 
ERNST   MACH,    Pn.D, 

PKOFESSOB  IN   THE   UNIVERSITY  OF   VIENNA, 

WHOSE  GENIUS  HAS  ILLUMINATED 

THE   HISTOEICAL  AND  PHILOSOPHICAL  DEVELOPEMENT   OF 
MECHANICS  AND  MANY  OTHER  BRANCHES   OF 

PHYSICAL  SCIENCE, 
THIS  BOOK  IS   GRATEFULLY   INSCRIBED. 


311873 


PEEFACE. 

TT  is  a  common  complaint  that  though  the  principles  of 
Mechanics  are  the  simplest  and  the  earliest  to  be  discovered  in 
the  whole  range  of  Science,  and  moreover  are  directly  illustrated 
in  almost  every  act  of  our  lives,  more  difficulty  is  found  in 
giving  beginners  a  real  grip  of  them  than  with  any  other  branch 
of  Physics. 

This  I  attribute  largely  to  the  way  in  which  the  text-books 
deal  with  the  subject.  The  student  usually  opens  the  book  upon 
a  chapter  in  which  such  leading  concepts  as  matter,  force,  mass, 
particle,  rigid  body,  smooth  body  are  treated  in  definitions  of 
a  line  or  two  each,  before  he  sees  any  reason  for  their  introduction 
at  all.  He  is  probably  warned  that  philosophers  are  not  agreed 
about  the  nature  of  matter  ;  that  motion  is  purely  relative ;  that 
force  is  a  misleading  idea  borrowed  from  our  muscular  sensations 
and  better  got  rid  of;  and  that  no  such  things  as  mathematical 
particles,  rigid  bodies  and  smooth  bodies  exist  in  nature.  He 
naturally  concludes  that  Mechanics  is  an  abstruse  subject  having 
nothing  to  do  with  realities  or  common  sense. 

The  second  chapter  plunges  him  into  the  mathematical  study 
of  motion  in  the  abstract.  Here  he  struggles  with  variable 
velocity  and  acceleration,  and  the  kinematic  formulae;  and  is 
lucky  if  he  is  let  off  without  a  discussion  of  motion  in  a  circle 
and  in  a  cycloid,  simple  harmonic  motion,  and  the  parabola. 
To  his  previous  confusion  he  adds  the  conviction  that  this  is  only 
another  branch  of  the  pure  mathematics  he  has  hitherto  found 
so  little  use  for. 


viii  PREFACE 

At  last  there  is  a  chapter  on  the  Laws  of  Motion,  so  in- 
adequately treated  that  he  oftens  ends  by  believing  that  they 
were  made  up  by  Sir  Isaac  Newton,  the  author,  so  far  as  he 
is  aware,  of  the  whole  subject.  The  rest  of  the  book  is  too  often 
merely  geometrical  and  trigonometrical  gymnastics. 

In  recent  years  many  text-book  writers  have  attempted  to 
break  away  from  this  mischievous  tradition.  Some  have  tried 
to  rewrite  the  whole  subject  from  the  latest  point  of  view  of 
Energetics.  But  this  is  surely  to  begin  at  the  wrong  end. 
According  to  the  biologists  the  bodily  development  of  the 
individual  is  an  epitome  of  the  development  of  the  race.  Is  not 
this  a  hint  that  the  historical  method  is  the  natural  way  of 
attacking  a  subject  of  study?  Others  have  sought  to  discard  the 
idea  of  force,  and  speak  only  of  mass-accelerations.  "Naturam 
expellas  furca."  It  is  rarely  indeed  that  they  manage  twenty 
pages  without  getting  back  to  the  old  point  of  view.  With 
proper  caution  the  use  of  this  concept  is  as  valuable  as  it  is 
historically  right  and  inevitable.  Still  others  have  set  the  student 
to  rediscover  the  subject  for  himself  by  experiment.  But  this 
wastes  too  much  time  on  mere  manipulation,  and  leaves  the 
student's  knowledge  in  mid  air,  unrelated  to  all  that  has  gone 
before  him  in  the  course  of  actual  discovery.  It  seems  a  pity  that 
he  should  close  the  book  without  a  glimmering  of  personal  interest 
in  his  predecessors,  the  great  investigators,  and  forego  the  insight 
into  philosophic  and  scientific  method  which  a  study  of  the 
development  of  Mechanics  evokes  insensibly  and  unawares. 

No  claim  for  originality  can  be  made  for  this  book.  And  yet 
I  find  it  difficult  to  make  detailed  acknowledgement  of  obligations, 
for  it  embodies  the  system  of  teaching  Mechanics  at  which  I  have 
arrived  after  thirty,  years'  experience,  and  it  is  no  longer  possible 
to  say  where  certain  illustrations  and  ways  of  putting  things  have 
come  from,  or  whether  (less  probably)  they  were  devised  by 


PREFACE  IX 

myself.     A  desire   to  shew  whence  the  general  plan  has  arisen 
must  be  my  excuse  for  a  personal  digression. 

After  learning  and  teaching  Mechanics  for  ten  years  on  the 
traditional  system  described  above,  I  was  called  on,  as  a  lecturer 
under  the  Cambridge  University  Extension  Scheme,  to  explain 
the  principles  to  audiences  without  any  previous  mathematical 
training,  but  often  composed  of  engineers,  plumbers,  and  other 
workmen  who  had  derived  excellent  practical  notions  on  the 
subject  from  their  experience.  Obliged  thus  to  recast  the  subject 
in  my  own  mind,  I  found  it  possible  to  present  all  the  main 
principles  with  the  aid  of  ordinary  arithmetic  and  the  simplest 
geometrical  diagrams.  At  this  stage  Sir  Robert  Ball's  admirable 
lectures  on  Experimental  Mechanics  gave  me  great  assistance. 
My  experience  with  these  popular  audiences  reacted  with  advan- 
tage on  my  teaching  with  classes  in  the  University,  and  fired  me 
with  the  ambition  to  write  a  text-book  on  Mechanics.  But  a 
sight  of  Sir  Oliver  Lodge's  excellent  Mechanics  in  Chambers' 
series  put  an  end  to  this  wish  for  a  time.  Some  ten  years  ago 
I  stumbled  on  the  first  German  edition  of  Professor  Mach's 
Die  Mechanik  in  ihre  Entwickelung.  I  am  ashamed  to  say 
that  this  fascinating  book  was  my  first  introduction  to  the 
historical  development  of  a  subject  I  had  taught  so  long.  Since 
then  my  teaching  has  been  based  more  and  more  on  the  lines  laid 
down  by  Mach,  and  as  I  have  found  it  impossible  to  induce 
ordinary  students  to  read  the  original,  even  when  translated, 
I  recurred  to  the  idea  of  writing  a  text-book  which  should  yet  be 
based  on  Mach's  method.  The  present  book  is  the  result.  In 
producing  it  I  have  kept  in  view  the  following  aims: — 

(1)  First  and  throughout,  to  make  a  text-book  of  mechanical 
principles,  avoiding  as  far  as  possible  merely  mathematical 
difficulties,  and  reserving  those  that  could  not  be  avoided  for 
separate  treatment  in  the  later  parts  of  the  book ;  but  never 
shirking  them  where  necessary. 


X  PREFACE 

(2)  To   develops   the    principles   in    their    historical    order, 
starting  from  real  problems,  as  the  subject  started,  shewing  how 
the  great  investigators  attacked  those  problems,  and  only  intro- 
ducing the  leading  concepts  as  they  arise  necessarily  and  naturally 
in  the  course  of  solving  them. 

(3)  To   bring    out    incidentally   the    points    of    philosophic 
interest  and  the  method  of  science. 

(4)  To  appeal  constantly  to  experiment,  as  far  as  possible  in 
the  original  form,  for  purposes  of  verification  in  the  early  part  of 
the  subject,  leading  up  to  an  experimental  course  limited  to  the 
most  important  practical  applications.     In  this  way  a  good  deal  is 
included  in  the  later  chapters  which  does  not  usually  find  a  place 
in  elementary  text-books. 

(5)  To  interest  the  student  in  the  personality  of  the  great 
pioneers,  and  if  possible  induce  the  habit  of  referring  to  original 
sources. 

(6)  Not  to  overload  the  text  with  masses  of  examples;   to 
give  only  so  many  that  every  student  should  work  them  all ;  and 
to  select  these  so  as  to  bring  in  useful  and  interesting  physical 
constants,  and  make  them  as  direct  as  possible,  discarding  all 
those   in   which   the   amount   of   mechanical   principle   involved 
is  a  mere   drop   to   an   intolerable   deal   of   pure   mathematical 
exercise.     Any  intelligent   teacher  can  multiply  examples   of  a 
given  type,  where  this  is  necessary,  either  from  his  own  invention 
or  from  the  numerous  extant  collections.     Unintelligent  teachers 
have  no  business  with  Mechanics. 

It  will  be  obvious  that  the  Book  could  never  have  been  written 
but  for  Mach's  Mechanik.  It  is  indeed  only  a  poor  and  incomplete 
abridgement  of  Mach's  work  intended  for  students.  I  trust  that 
every  teacher  into  whose  hands  it  may  fall,  and  many  students, 
will  be  driven  by  it  to  the  original.  It  is  impossible  for  me 
to  express  my  personal  obligation  to  Professor  Mach  in  connection 
with  this  subject,  or  the  respect  and  gratitude  I  feel  for  such 


PREFACE  XI 

a  master  for  enlightenment  and  inspiration  in  this  as  in  many 
other  branches  of  Physics. 

Further  acknowledgement  must  be  made  of  help  from  Clerk 
Maxwell's  Matter  and  Motion,  Sir  Oliver  Lodge's  Pioneers  of 
Science,  Mr  W.  W.  R.  Ball's  Essay  on  Newton  s  Principia,  Professor 
Wright's  Mechanics,  and  Glazebrook  and  Shaw's  Practical  Physics, 
to  which  many  of  the  experiments  described  in  the  latter  part  of 
the  book  are  due.  The  Dynamics  of  Principal  Garnett  must  be 
specially  mentioned,  for  it  was  in  reading  the  proof-sheets  of  that 
work  that  I  first  learned  to  connect  the  familiar  formulae  with 
practical  facts. 

My  especial  thanks  are  due  to  Mr  F.  H.  Neville,  one  of  the 
Editors  of  the  Cambridge  Physical  Series,  not  only  for  his  extreme 
care  in  revising  the  book  for  the  press,  but  for  many  most  valuable 
criticisms  and  suggestions  that  have  led  to  important  improve- 
ments. 

The  course,  as  laid  down,  has  been  tested  with  classes  at 
McGill  University,  and  has,  I  think,  proved  interesting  and 
intelligible.  Many  of  the  illustrations  are  from  photographs 
of  actual  apparatus  used  here.  Whether  the  book  will  be  found 
adapted  to  students  anywhere  else  is  a  subject  of  much  misgiving. 
It  aims  at  no  particular  examinations,  and  may  be  too  revolutionary 
to  find  favour  in  schools.  Its  execution  is  probably  faulty  enough. 
But  that  something  of  this  nature  would  be  a  more  worthy 
treatment  of  the  subject  for  university  students,  even  for  the 
ordinary  degree,  than  the  present  jejune  versions  of  Varignon's 
Statics  and  mathematical  exercises  on  kinematics,  I  have  no  doubt 
at  all.  Until  Mechanics  is  clad  in  its  historical  flesh  and  blood, 
it  will  remain  the  dull  and  tiresome  subject  that  has  convinced 
so  many  generations  of  students  that  an  abysmal  gulf  separates 

theory  from  practice. 

J.  C. 
MONTREAL, 

April,  1904. 


CONTENTS. 


BOOK   I 

THE  WINNING  OF  THE   PRINCIPLES. 

CHAPTER  PAGE 

Introduction       ..........  1 

I.  The    Beginnings   of  Statics.     Archimedes.     Problem   of  the 

Lever  and  of  the  Centre  of  Gravity          ....  3 

II.  Experimental  Verification  and  Applications  of  the  Principle 

of  the  Lever        .........  10 

III.  The   Centre  of  Gravity 23 

IV.  The  Balance 32 

V.  Stevinus  of  Bruges.     The  Principle  of  the  Inclined  Plane     .  41 

VI.  The  Parallelogram  of  Forces     .         .        .         .         .        .         .48 

VII.  The  Principle  of  Virtual  Work 52 

VIII.  Review  of  the  Principles  of  Statics 66 

IX.  The  Beginnings  of  Dynamics.   Galileo.    The  Problem  of  Falling 

Bodies 69 

X.  Huyghens.     The   Problem  of  Uniform   Motion   in  a  Circle. 

"Centrifugal  Force" 84 

XL  Final  Statement  of  the  Principles  of  Dynamics.  Extension  to 
the  Motions  of  the  Heavenly  Bodies.  The  Law  of  Uni- 
versal Gravitation.  Newton  88 


BOOK   II. 

MATHEMATICAL  STATEMENT  OF  THE   PRINCIPLES. 

Introduction       .         .         .         .         ...         .         .         .         .  103 

XI  I.      Kinematics         ..........  105 

XIIL     Kinetics  of  a  Particle  moving  in  a  Straight  Line.     The  Laws 

of  Motion 116 

XIV.     Experimental  Verification  of  the  Laws  of  Motion.     Atwood's 

Machine 136 


XIV 


CONTENTS 


XV. 
XVI. 
XVII. 

XVIII. 
XIX. 


Work  and  Energy 

The  Parallelogram  Law         .... 
The  Composition  and   Resolution  of    Forces. 

Component.     Equilibrium 
Forces  in  One  Plane 


Resultant. 


142 
155 

159 
173 
Friction  .  193 


BOOK   III. 

APPLICATION  TO  VARIOUS  PROBLEMS. 

XX.  Motion  on  an  Inclined  Plane.     Brachistochrones. 

XXI.  Projectiles 

XXII.  Simple  Harmonic  Motion      .        .        . 

XXIII.  The  Simple  Pendulum 

XXIV.  Central  Forces.     The  Law  of  Gravitation    . 

XXV.  Impact  and  Impulsive  Forces       .... 


207 
213 
224 
237 
243 
257 


BOOK   IV. 

THE  ELEMENTS  OF  RIGID  DYNAMICS. 

XXVI.  The  Compound  Pendulum.     Huyghens3  Solution         .        .271 

XXVII.  D'Alembert's  Principle 2*76 

XXVIII.     Moment  of  Inertia 281 

XXIX.  Experimental  Determination  of  Moments  of  Inertia            .  289 

XXX.  Determination  of  the  Value  of  Gravity  by  Eater's  Pendulum  303 

XXXI.  The  Constant  of  Gravitation,  or  Weighing  the  Earth.     The 

Cavendish  Experiment 313 

ANSWERS  TO  THE  EXAMPLES 324 

INDEX  328 


PLATES. 


Archimedes 
Galileo      . 
Huyghens 
Newton    . 


to  face  page       69 


85 
89 


BOOK   I. 

THE    WINNING   OF    THE    PRINCIPLES. 


OF  THE 
UNIVERSITY 

OF 


INTRODUCTION. 

BY  Mechanics  is  understood  nowadays  the  science,  or  organized 
body  of  knowledge  we  possess  concerning  the  conditions  of  rest  or 
motion  of  the  objects  about  us. 

How  did  we  come  by  it  ?  The  word  itself  means  "  con- 
trivances," and  gives  a  hint  that  the  science  arose  from  the 
devices  which  were  found  helpful  in  lifting  weights  and  moving 
objects  to  satisfy  practical  needs.  Long  before  there  was  any 
collection  of  rules,  much  less  a  science  of  Mechanics,  the  ad- 
vantages of  the  traditional  "  Mechanical  Powers  " — the  Lever ;  the 
Wheel  and  Axle  (or  continuous  lever);  the  Pulley  (or  travelling 
lever) ;  the  Inclined  Plane  ;  the  Wedge  (or  double  inclined  plane) ; 
and  the  Screw  (or  continuous  inclined  plane) — were  known.  The 
great  monuments  of  antiquity,  like  the  Pyramids,  could  hardly 
have  been  raised  by  the  labour  of  unaided  hands.  As  a  matter  of 
fact,  rude  implements  of  the  kind  have  been  found  in  ancient 
graves,  and  the  Egyptian  and  Assyrian  records  contain  pictorial 
representations  of  such  appliances. 

The  transition  to  what  may  be  properly  called  science  takes 
place  when,  for  example,  instead  of  the  practical  knowledge  that  a 
great  weight  may  be  lifted  by  a  small  one  with  the  aid  of  a  crow- 
bar, a  principle,  or  rule,  is  discovered,  which  tells  us  what  must  be 
the  lengths  of  the  arms  of  the  crowbar  in  order  that  a  certain 
small  weight  may  lift  a  given  large  weight.  This  is  a  step  of 
immense  importance.  For  once  it  is  made,  a  craftsman  can  save 
innumerable  mistakes,  with  their  consequent  loss  of  time  and  risk 
of  injury,  by  calculating  beforehand  what  weight  or  length  of 
lever  to  employ ;  and  he  can  communicate  to  others  what  he  has 
learned  from  his  own  experience.  Thus  both  Design  and  the 
c.  1 


2  MECHANICS 

Dissemination  of  Knowledge  become  possible,  as  when,  during  the 
siege  of  Kimberley,  a  mining  engineer  constructed  out  of  a  steel 
axle,  ten  feet  long,  a  gun  capable  of  replying  to  the  Boer  artillery, 
gathering  his  information  out  of  some  back  numbers  of  the 
engineering  magazines,  which  happened  to  be  in  his  possession. 

Statics,  the  part  of  the  subject  which  deals  with  Equilibrium, 
being  simpler  and  far  more  directly  concerned  with  the  Mechanical 
Powers,  took  its  rise  much  earlier  than  Dynamics,  the  science  of 
motion  as  produced  by  force.  It  begins  with  the  clearing  up  of 
the  Principle  of  the  Lever  by  Archimedes,  to  whom  is  also  due  the 
fundamental  principle  of  Hydrostatics.  Strange  as  it  may  seem, 
these  are  the  only  important  contributions  to  Physical  Science  of 
the  ancient  world  Tip  to  the  middle  of  the  sixteenth  century,  when, 
after  an  interval  of  eighteen  hundred  years,  Stevinus  of  Bruges 
attacked  the  study  of  Statics  again,  this  time  by  way  of  the 
Inclined  Plane. 

From  this  time  onwards  a  series  of  great  investigators, — 
Galileo  (1564—1642),  Huyghens  (1629—1695),  and  Newton 
(1642 — 1727) — laid  the  foundations  of  the  Dynamics  of  a  single 
particle,  and  bodies  that  could  be  treated  as  such.  D'Alembert 
(1743)  gave  a  general  principle  by  which  Newton's  ideas  could 
be  applied  to  the  complicated  case  of  solid  bodies  consisting 
of  innumerable  particles. 

All  that  remained  to  be  done  was  to  employ  the  highest 
developments  of  Pure  Mathematics  in  working  out  the  con- 
sequences of  the  principles  already  discovered ;  save  that  towards 
the  middle  of  the  nineteenth  century  the  importance  of  the 
doctrine  of  Energy  came  to  be  more  and  more  recognized,  and 
the  great  generalization  known  as  the  Conservation  of  Energy, 
dimly  foreshadowed  in  the  Principle  of  Virtual  Work  and  the 
Scholia  to  Newton's  Third  Law  of  Motion,  was  finally  established 
and  extended  from  Mechanics  to  all  departments  of  Physics. 

We  shall  begin  with  Archimedes  and  the  Principle  of  the 
Lever.  It  is  instructive  to  examine  this  case  in  some  detail,  not 
only  for  its  historical  interest,  but  because  it  is  an  admirable 
example  of  the  way  in  which  Physical  Science  has  developed. 


CHAPTER  I. 

ARCHIMEDES—  PEOBLEM  OF  THE  LEVER  AND  OF 
THE  CENTRE  OF  GRAVITY. 


Aos   TTOV  crra>,   KOI  TTjv  yrjv 

1.  ARCHIMEDES  (287  —  212  B.C.),  the  greatest  mathematical 
and  inventive  genius  of  antiquity,  was  born  at  Syracuse,  and  com- 
pleted his  education  at  Alexandria  under  Conon,  in  the  Royal 
Schools  of  the  Ptolemies,  of  which  Euclid  had  been  an  ornament 
fifty  years  earlier.     The  stories  of  the  crown  of  Hiero,  the  burning 
mirrors,  and  his  slaughter  at  the  end  of  the  siege  in  spite  of 
Marcellus'  orders  are  well  known. 

2.  Every  one  knows  that  a  small  force  applied  to  one  end  of 
a  lever,  a  long  way  from  the  point  of  support,  or  fulcrum,  will 
overpower  a  much  larger  force  applied  near   the  fulcrum.    The 
problem  is  to  find  a  rule  connecting  the  forces  and  their  distances 
from  the  fulcrum  when  they  just  balance. 

Those  who  consult  the  treatise  of  Archimedes  rrepl  eVt7re8o>z/ 
IcroppOTrifcwv  rj  /cevrpa  ffapcov  eTrnreScov,  will  be  struck  by  two 
things.  He  deals  entirely  with  weights,  not  introducing  the 
general  notion  of  force;  and  he  is  so  deeply  imbued  with  the 
methods  of  the  ancient  geometers  that  he  tries  to  cast  his  proofs 
of  physical  propositions  into  the  same  form.  His  very  real  grip 
of  the  Principle  of  the  Lever  must  have  been,  in  his  case  as  well 
as  that  of  the  craftsmen  of  his  age,  a  direct  result  of  experience. 
Yet  he  finds  a  satisfaction  in  reducing  it  to  the  already  familiar 
array  of  Axioms,  Proofs  by  Reductio  ad  Absurdum,  and  the 
geometrical  theory  of  Proportion. 

1—2 


MECHANICS 


[CHAP. 


He  begins  by  laying  down  the  following  Axioms : 

(1)  Equal  weights  placed  at  equal  distances  from  the  point  of 
support  balance. 

(2)  Equal  weights  placed  at  unequal  distances  do  not  balance, 
but  that  which  hangs  at  the  greater  distance  descends. 

Then  follows  a  proof  by  reductio  ad  absurdum  that  in  the  case 
of  unequal  weights  balancing  at  unequal  distances,  the  greater 
weight  must  be  at  the  shorter  distance.  Before  advancing  to  the 
actual  numerical  law  connecting  the  weights  and  the  distances,  he 
now  lays  down  three  propositions  to  shew  that  the  Centre  of 
Gravity  of  any  number  of  equal  weights,  odd  or  even,  equally 
spaced  out  along  a  bar,  must  be  at  the  middle  point  of  the  bar. 
Observe  that  in  these  propositions  it  is  clear  that  he  conceives  of 
the  centre  of  gravity  as  a  point  such  that,  if  it  be  supported,  the 
weights  will  balance  about  it. 

He  is  now  able  to  give  a  beautifully  ingenious  proof  of  the 
general  principle  of  the  Lever,  viz.  that  for  equilibrium  the  weights 
must  be  inversely  proportional  to  the  distances ;  first  for  the  case 
of  commensurable,  and  then  for  incommensurable  weights.  As  a 
historical  curiosity,  we  shall  give  the  former  in  his  own  words. 

3.  Proposition.  Magnitudes  whose  weights  are  commensur- 
able will  balance  if  they  are  hung  at  distances  which  are  inversely 
proportional  to  their  weights. 


Fig.  1. 


Let  a,  6,  be  commensurable  weights.     Let  ed  be  any  distance, 
and  let  dc  be  to  ce  as  a  is  to  b.     It  has  to  be  proved  that  the 


l]  ARCHIMEDES  5 

centre  of  gravity  of  the  magnitude  composed  of  both  a  and  6, 
placed  at  e  and  d  respectively,  is  the  point  c. 

Since  dc  is  to  ce  as  a  is  to  b,  and  a  is  commensurable  with  b, 

.'.  dc  is  commensurable  with  ce,  a  straight  line  with  a 
straight  line. 

.'.  there  must  be  some  common  measure  of  dc,  ce. 

Let  it  be  n ;  and  take  dg,  dk,  on  each  side  of  d,  equal  to  ce, 
and  el  equal  to  dc. 

Since  dg  —  ec, 

.'.  also  cfc  =  6(7, 

/.  also  le  =  eg, 

.' .  Ig  is  double  of  c?c,  and  gk  of  ec. 

/.  n  will  measure  both  Z#  and  gk,  since  it  measures  their  halves. 

And  since  dc  is  to  ce,  as  a  is  to  b,  and  Z#  is  to  gk,  as  cfc  is  to  ce, 
for  they  are  the  double  of  each,  .*.  Ig  is  to  gk  as  a  is  to  6. 

Now  let  a  be  the  same  multiple  of  a  magnitude  /,  that  Ig 
is  of  n. 

Then  a  is  to /"as  Ig  is  to  n. 

But  kg  is  to  Jgr  as  6  is  to  a. 

/.  ex  aequali,  b  is  toy  as  kg  is  to  n. 

.*.  kg  is  the  same  multiple  of  n  that  6  is  off. 

But  it  has  been  shewn  that  a  is  also  a  multiple  of/ 

.*.  /is  a  common  measure  of  a  and  6. 

If  therefore  Ig  be  divided  into  parts  equal  to  n,  and  a  irito 
parts  equal  to/  the  parts  of  £0r,  each  equal  to  n,  will  be  the  same 
in  number  as  the  parts  of  a,  each  equal  to/ 

.'.  if  to  each  of  the  parts  of  Ig  there  be  applied  a  magnitude 
equal  to/  having  its  centre  of  gravity  in  the  middle  of  the  part, 
all  the  magnitudes  will  together  be  equal  to  a,  and  the  centre  of 
gravity  of  the  magnitude  composed  of  all  of  them  will  be  e,  for 
they  are  equal  in  number  on  opposite  sides,  since  le  =  eg. 

Similarly  it  can  be  shewn  that  if  to  each  of  the  parts  of  kg 
there  be  applied  a  magnitude  equal  to  /  having  its  centre  of 
gravity  in  the  middle  of  the  part,  all  the  magnitudes  will  together 
be  equal  to  b,  and  the  centre  of  gravity  of  the  magnitude  composed 
of  them  all  will  be  the  point  d. 

a  has  therefore  been  placed  at  e,  and  b  at  d,  and  there  are  now 


MECHANICS 


CHAP. 


an  even  number  of  equal  magnitudes,  placed  in  a  straight  line, 
whose  centres  of  gravity  are  equally  distant  from  each  other. 

It  follows  that  the  centre  of  gravity  of  all  the  magnitudes 
together  is  the  point  of  bisection  of  the  straight  line  in  which  the 
centres  of  gravity  of  the  magnitudes  lie. 

.*.  since  le  =  cd,  and  ec  =  dk,  the  whole  Ic  —  the  whole  ck. 

/.  the  centre  of  gravity  of  the  whole  is  the  point  c. 

.*.  if  a  be  placed  at  e,  and  6  be  placed  at  d,  they  will  balance 
about  the  point  c.  Q.  E.  D. 

4.  But  little  advance  was  made  on  this  cumbrous  proof  for 
1800  years,  when  Stevinus  of  Bruges  (1548 — 1620  A.D.)  gave  it 
the  following  interesting  form. 


? 

*                         N 

E                E 

K 

1 

L 

M 

Fig.  2. 

Consider  a  uniform  column,  AC,  suspended  by  its  middle  point 
M,  so  that  it  will  balance.  Imagine  it  divided  at  EF  into  two 
parts  whose  middle  points  will  be  K  and  L.  Then  the  weights  of 
AF,  EC  are  proportional  to  GI  and  IH,  and  may  be  supposed 
collected  at  K  and  L. 

It  is  easily  seen  that  KM  =  IL  and  ML  =  IK. 

.'.  the  greater  weight  AF  is  to  the  smaller  weight  EC  as  the 
longer  arm  ML  is  to  the  shorter  arm  MK. 

The  second  figure  shews  that  the  weights  may  be  hung  at  any 
depth  below  the  bar,  and  that  any  equal  weights  may  be  sub- 
stituted for  the  parts  of  the  bar. 

This  form  of  the  proof  was  adopted,  with  a  slight  modification, 
by  Galileo  (1638)  and  in  modern  times  by  Lagrange. 


l]  STEVINUS.      THE    LEVER  7 

5.  In  this  proof  Archimedes  and  his  successors  apparently 
criticism  of  evolve  a  physical  truth  by  geometrical  methods 
the  proof.  from  certain  axioms  which  are  assumed  as  self- 

evident  apart  from  experience.     But  can  this  be  possible  ? 

Take  the  first  axiom.  It  seems  perhaps  self-evident  that  equal 
weights  at  equal  distances  from  the  fulcrum  must  balance,  from 
the  mere  symmetry  of  the  figure.  The  ancient  philosophers, 
steeped  in  the  methods  of  logic  and  geometry,  base  such  cases  of 
symmetry  on  what  was  called  the  "Principle  of  Sufficient  Reason." 
No  motion  can  take  place,  because  there  is  no  reason  why  the 
balance  should  descend  on  one  side  more  than  the  other.  "  But 
we  forget  in  this  that  a  great  multitude  of  negative  and  positive 
experiences  is  implicitly  contained  in  our  assumption ;  the  negative, 
for  instance,  that  unlike  colours  of  the  lever  arms,  the  position  of 
the  spectator,  an  occurrence  in  the  vicinity,  and  the  like,  exercise 
no  influence ;  the  positive,  on  the  other  hand  (as  it  appears  in  the 
second  axiom),  that  not  only  the  weights,  but  also  their  distances 
from  the  supporting  point  are  decisive  factors  in  the  disturbance 
of  equilibrium  "  (Mach,  The  Science  of  Mechanics). 

The  secret  of  the  immense  and  rapid  development  of  natural 
knowledge  in  modern  times  lies  in  the  deliberate  and  faithful 
ransacking  of  nature  for  her  facts,  since  the  time  of  Francis 
Bacon's  Novum  Organon.  Natural  processes  can  only  be  learned 
from  experience ;  they  cannot  be  extracted  from  the  meanings  of 
words  or  the  canons  of  logic,  after  the  manner  of  the  ancient 
world,  except  in  so  far  as  these  themselves  enshrine  the  results  of 
direct  experience,  hereditary  or  personal.  Why  should  not  the 
position  of  the  sun  affect  the  balancing  of  an  equal-armed,  equally- 
weighted  lever,  so  that  it  should  be  horizontal  at  noon  and  mid- 
night, and  its  eastern  limb  dip  in  the  morning,  its  western  in  the 
evening  ?  Nothing  but  experiment  can  teach  us  that  the  sun  has 
no  effect  in  this  case. 

Let  a  wire  pointing  to  magnetic  north  be  stretched  horizontally 
over  a  compass  needle,  and  let  an  electric  current  be  sent  through 
the  wire  from  South  to  North.  Everything  is  symmetrical.  It 
might  seem  an  axiom  in  accordance  with  the  principle  of  sufficient 
reason  that  the  magnet  will  remain  at  rest.  Yet  Oerstedt  dis- 
covered in  1821  that  the  north  pole  of  the  magnet  will  certainly 
turn  to  the  west. 


8  MECHANICS  [CHAP. 

In  the  Proposition  Archimedes  is  seeking  to  reduce  the  general 
and  more  unfamiliar  case  of  a  lever  with  unequal  arms  to  the  case 
of  the  equal-armed  lever,  which  was  already  so  familiar  to  him 
that  the  knowledge  of  it  seemed  instinctive  or  axiomatic.  Think 
of  any  case  of  scientific  explanation,  and  you  will  see  that  this  is 
all  that  is  accomplished, — the  reduction  of  unfamiliar  cases  to 
those  already  familiar.  Newton  discovers  that  the  moon  in  her 
orbit  drops  towards  the  earth  according  to  the  same  law  that  is 
familiar  in  the  falling  apple  ;  the  motion  of  the  moon  is  explained, 
but  not  that  of  the  apple. 

But  did  Archimedes  succeed  ?  It  seems  unlikely  that  a  method 
which,  apart  from  experience,  fails  to  justify  his  first  axiom,  can 
possibly  lead  him  to  the  numerical  law  of  the  lever.  Closely 
scrutinized,  the  fallacy  appears.  He  assumes  that  a  number  of 
weights  spaced  out  along  one  arm  of  a  lever  will  have  the  same 
turning  effect  about  the  fulcrum,  as  if  they  were  all  collected  at 
their  centre  of  gravity ;  whereas  what  he  has  proved  from  his 
axioms  in  the  preliminary  propositions  is  that  they  will  balance 
about  their  centre  of  gravity,  if  it  be  supported. 


Fig.  3. 

Suppose  it  had  been  a  question,  not  of  balancing,  but  of  the 
resistance  experienced  upon  attempting  to  set  the  lever  AB 
rotating  rapidly  about  C.  We  require  to  find  the  law  connecting 
unequal  weights  at  unequal  distances,  so  that  they  may  offer  the 
same  resistance.  May  we  substitute  for  A,  two  weights  each 
equal  to  J./2,  placed  symmetrically  about  A  ?  Certainly  not. 
But  it  is  only  experience  that  tells  us  we  may  do  in  a  question 
of  balancing,  or  centres  of  gravity,  what  we  may  not  do  when  it 
concerns  moments  of  inertia  and  rotation.  But  Archimedes  is 
not  aware  that  he  has  made  the  step,  because  he  has  been  busied 
with  the  principle  of  the  centre  of  gravity,  which  is  in  fact 
equivalent  to  the  principle  of  the  lever.  He  would  not  have 


l]  THE    LEVER  9 

attempted  the  proof,  if  he  had  not  first  discerned  the  principle 
directly,  and  the  fact  of  experience,  once  discerned,  has  as  great 
an  authority  in  the  general  case,  as  in  the  simple. 

But  the  achievement  of  Archimedes  is  not  in  vain,  for  it 
brings  into  vivid  relief  the  connection  between  the  general  case  of 
unequal  arms,  and  the  special  and  more  familiar  instance,  when 
the  arms  are  equal ;  and  we  derive  a  satisfaction  from  our  insight 
into  their  consistency. 


CHAPTER   II. 

EXPERIMENTAL  VERIFICATION  AND  APPLICATIONS  OF 
THE   PRINCIPLE  OF  THE   LEVER, 

6.  WHAT   the   early   investigators   learned    from    their   own 
experience  and  that  of  the  craftsmen,  the  student  of  to-day  can 
only  grasp  with  equal  vividness  by  experimenting  for  himself. 

The  Principle  to  be  verified  is  this : — 

Let  AB  be  a  light  rod  supported  at  C,  and  let  weights,  P  and 
Q,  be  hung  at  A  and  B.     Then  for  equilibrium 

P:Q::BC:AC, 
or  more  conveniently, 

Px  AC=QxBC, 

i.e.,  the  products  of  the  weights  by  the  arms  at  which  they  hang 
are  the  same. 

7.  Experiment  1.     Take  a  graduated  rod,  say  a  30-centimeter 
length  of  a  meter  rod,  from  which  two  scale-pans  can  be  supported 
by  loops  of  fine  wire.     Suppose  the  weight  of  each  scale-pan  is 
made   up  to  50  grams  by  adding  lead  shot.     Place  a  50-gram 
weight  in  each,  and  balance  the  rod  over  any  sharp  edge,  such  as 
a  small  metal  or  glass  prism,  on  a  corner  of  a  table,  so  as  to  allow 
the  scale-pans  to  hang  below,  one  at  each  end,  and  each  of  them 
15  cm.  from  the  prism  at  the  centre.     Keeping  one  of  the  scales 
unchanged,  find  where  the  other  scale  must  hang  in  order  to 
balance  about  the  centre,  when  the  weight  in  it  is  increased  by 
20  gms.  at  a  time  up  to  a  total  weight  of  250  gins. 


CHAP.  II]  THE    PRINCIPLE   OF   THE    LEVER 

Make  a  table  as  follows  :  — 
100  gms.  (including  scale)  balance  100  gins,  at 


11 


Products 


140 


„        250         „ 

«!,  a2,  &c.  being  the  observed  distances  from  the  centre  of  the  rod 
at  which  the  second  scale  has  to  be  hung  to  secure  a  balance. 

The  rod  may  not  be  quite  uniform,  so  that  c^  may  not  be 
exadtly  15  cms.  But  work  out  the  products  of  the  weights  and 
distances  on  the  right-hand  side,  and  see  whether  they  are  as  nearly 
equal  as  the  accuracy  of  your  method  would  reasonably  lead  you 
to  expect. 

8.  Stevinus,  as  Archimedes  1800  years  before  him,  is  thinking 
always  of  real  weights.  Even  when  the  pull  is  to  be  exerted 
upwards,  as  in  supporting  the  rod  at  (7,  it  is  applied,  as  his  figures 


Fig.  4.     (From  Stevinus.) 

shew,  by  means  of  a  string  carried  over  a  fixed  wheel,  or  pulley, 
with  the  proper  weight  hanging  from  the  other  end. 


12  MECHANICS  [CHAP. 

If  R  is  this  sustaining  weight,  we  may  verify  that 

R  =  P  +  Q. 

Experiment  2.  But  it  is  more  convenient  to  employ  a  Spring 
Balance,  which  determines  weights  by  the  amount  to  which  they 
can  pull  out  a  spiral  spring  fastened  at  one  end. 

Attach  such  a  spring  balance  to  the  centre  of  the  rod,  having 
first  verified  the  readings  of  the  balance  by  testing  it  with  a  set  of 
standard  weights,  and  made  a  table  of  errors. 

o 


Fig.  5. 

Repeat  several  of  the  above  experiments,  noting  the  total  weight 
supported.  Weigh  the  rod  without  the  scale-pans,  and  subtract 
its  weight  from  each  of  the  total  weights.  See  if  the  remainders 
are  not  in  every  case  equal  to  the  sum  of  the  weights  suspended 
from  the  rod. 

9.  Experiment  3.     Attach  a  small    object,  such  as  a  metal 
clamp,  to  one  end  of  the  rod,  and  balance  it  without  scale-pans. 
The  point  over  the  edge  of  the  prism  must  be  the  centre  of  gravity 
of  the  rod  and  clamp  together,  and  their  joint  weight  may  be 
supposed  to  act  at  it. 

Hang  a  scale-pan  with  a  known  weight  near  the  other  end, 
and  find  where  the  whole  balances.  Hence  calculate  the  weight 
of  the  rod  and  clamp  together,  and  verify  your  result  on  the  spring 
balance. 

10.  In  these  experiments  any  of  the  weights  concerned  may 

be  replaced  by  a  pull  or  push  applied  either  by  a 

spring  balance,  or  direct  muscular  effort,  which 

would  just   sustain    the   weight.     In    such    etforts  we   speak   of 


II]  LEONARDO    DA   VINCI.      THE    LEVER 

exerting  a  force,  and  it  is  convenient  to  introduce  this  term  at 
once,  though  the  general  idea  of  Force,  as  anything  whicli  changes 
or  tends  to  change  a  body's  state  of  rest  or  motion,  belongs  to 
Newton's  time,  half  a  century  after  Stevinus. 

11.     Our   experiments   shew   that   if  two    forces  be  applied, 
perpendicularly,  to  the  ends  of  a  rod  capable  of 

Moment  of  a  ... 

Force  about  a  turning  about  a  fixed  point,  or  fulcrum,  they  will 
balance  provided  the  product  of  each  force  into 
the  lever  arm  at  which  it  acts  is  the  same  for  both.  In  the  case 
we  have  considered  the  lever  arms  are  the  shortest,  or  perpen- 
dicular, distances  from  the  fulcrum  to  the  lines  of  action  of  the 
forces. 

Leonardo  da  Yinci  (1452 — 1519),  the  famous  painter,  engineer 
and  investigator,  recognized  that  this  is  the  essential  condition  in 
all  cases,  even  when  the  forces  act  obliquely.  He  says,  for  ex- 
ample :  We  have  a  bar  AD  free  to  rotate  about  A,  and  suspended 


Fig.  6. 

from  the  bar  a  weight  P,  and  suspended  from  a  string  which 
passes  over  a  pulley,  a  second  weight  Q.  What  must  be  the 
ratio  of  the  forces  that  equilibrium  may  obtain  ?  The  lever  arm 
for  the  weight  P  is  not  AD,  but  the  "  potential "  lever  AB.  The 
lever  arm  for  the  weight  Q  is  not  AD,  but  the  "potential"  lever 
AC. 

Professor  Mach  suggests  that  Leonardo  may  have  reached  this 
idea  in  some  such  way  as  this.     Consider  a  string  laid  round  a 


MECHANICS 


[CHAP, 


pulley,  and  subject   to   equal  tensions   on   both    sides.     EF  will 

be  a  plane  of  symmetry,  and      _ 

we    see   that   equilibrium  will 

subsist.       But    we    also    note 

that  the  only  essential  parts  of 

the   pulley  are  the   two  rigid 

radii,  AB,  AC,  which  determine 

the  form  of  the  motion  of  the 

points  of  application  of  the  two 

strings.     If  nails  were  driven 

through  the  string  at  B  and 

G,  the  rest  might  be  cut  away 

without  disturbing  equilibrium. 

Hence,  in  Fig.  6,  the  lever  arm 

for  the  right-hand  force  is  not 

AD,  but  the  "  potential "  lever 

AC. 


F 


Fig.  7. 


However  this  may  be,  it  was  recognized  that  the  torque,  or 
tendency  of  a  force  to  turn  a  body  about  a  pivot,  depended  only 
on  two  things,  the  magnitude  of  the  force,  and  the  perpendicular 
distance  from  the  pivot  to  its  line  of  action,  and  that  two  forces 
had  equal  torques,  if  for  each  the  product  of  the  force  by  the 
perpendicular  distance  of  its  line  of  action  from  the  fulcrum  was 
the  same.  This  product  is  therefore  the  measure  of  the  torque, 
or  tendency  of  a  force  to  turn  a  body  about  a  pivot. 

It  is  evidently  high  time  to  introduce  a  single  word  for  the 
very  important  but  cumbrous  expression  "  product  of  a  force  into 
the  perpendicular  distance  from  the  point  to  the  line  of  action 
of  the  force."  It  is  called  the  Moment  of  the  force  about  the 
point. 

Definition.  The  Moment  of  a  Force  about  a  point  is  the 
product  of  the  Force  by  the  perpendicular  distance  of  the  point 
from  the  line  of  action  of  the  force. 

We  can  now  state  the  Principle  of  the  Lever,  including  the 
case  of  oblique  forces,  as  follows: 

Two  forces  acting  on  a  lever  will  balance  when  their  Moments 
about  the  fulcrum  are  equal  and  opposite. 


n] 


MECHANICAL   ADVANTAGE 


15 


Mechanical 
Advantage. 


12.  The    Lever    and    the    other    Mechanical    Powers    were 

employed  to  enable  a  small  force  to  balance  or 
overcome  a  large  weight  or  force.  In  this  they 
are  said  to  aiford  "Mechanical  Advantage."  The  mechanical 
advantage  is  measured  by  the  ratio  of  the  large  weight  to  the 
force  required  to  balance  it.  Tradition  has  fixed  the  use  of  the 
terms  Power  and  Weight  to  indicate  the  force  employed  and  the 
resistance,  whether  weight,  or  pull,  or  push,  overcome.  This  is 
rather  unfortunate,  as  Power  has  a  definite  and  quite  different 
meaning  in  Dynamics. 

The  mechanical  advantage  of  the  Lever,  then, 
Weight  _  Length  of  Power  arm 
Power  ~~  Length  of  Weight  arm ' 

In  the  machines  the  weight  moved  is  not  always  greater  than 
the  power.  When  it  is  less,  the  power  is  said  to  act  at  mechanical 
disadvantage. 

13.  Consider  a  lever  with  arms  a,  6,  weights  P,  Q,  and  sup- 
The  different         ported  at  the  fulcrum  by  a  force  P  +  Q  applied 
kinds  of  levers.       by  ft  gpring  Balance.     It  does  not  matter  how  the 

three  forces  are  applied  at  A,  B,  (7,  provided  they  have  the  proper 

P+Q 


G 


Fig.  8. 

magnitudes.     We   may  therefore   regard  either  A   or  B  as  the 
fulcrum,  just  as  well  as  C. 

(1)  If  the  fulcrum  is  at  C,  between  the  power  and  the  weight, 
the  lever  is  said  to  be  of  the  first  class,  and  there  will  be  mechanical 
advantage  or  disadvantage  according  as  a  is  greater  or  less  than  6, 
since  P  x  a  =  Q  x  b. 


16 


MECHANICS 


[CHAP. 


Fig.  9.  Buckton  150-ton  Testing  Engine  in  the  Laboratories  of  McGill 
University,  Montreal.  The  V-supports,  of  which  there  are  two  sets  for  use  with 
different  scales,  are  seen  at  A.  The  beam  is  balanced  when  the  2000  Ib.  weight  B 
is  at  the  extreme  right.  The  specimen  to  be  tested  is  held  in  jaws  at  C.  The 
pull  is  applied  to  the  lower  end  by  hydraulic  machinery  in  the  room  below,  and 
the  weight  B  is  shifted  by  the  gear-wheel  D  so  as  to  keep  the  beam  balanced.  The 
tension  at  which  the  specimen  yields,  as  well  as  continuously  throughout  the 
operation,  is  read  on  the  scale. 


II]  EXAMPLES   OF   LEVERS  17 

Examples  of  levers  of  the  first  class  are : — Single  levers.  A 
poker  (lifting  the  coals  by  resting  on  the  bars  of  the  grate  as  a 
fulcrum);  a  crowbar;  the  shadoof,  or  pole  and  bucket,  a  device 
used  in  Egypt  for  raising  water  from  the  Nile ;  a  Testing  Engine. 

Double  levers.     A  pair  of  scissors ;  a  pair  of  pincers. 

(2)  Regard  B  as  the  fulcrum,  and  P  +  Q  as  the  "  weight "  ; 
the  power  P  is  on  the  same  side  of  the  fulcrum,  but  farther  off. 
The  lever  is  of  the  second  class,  and  there  is  always  mechanical 
advantage.     The  principle  of  the  lever  still  holds  good,  for 

(P  +  Q)xBC  =  PxBC+QxBC 
=PxBC+PxAG 
=  Px(BC  +  AC) 
=  PxAB, 

i.e.  product  of  power  and  its  arm  =  product  of  weight  and  its  arm. 

Examples  of  levers  of  this  class  are : — Single  levers.  The  oar 
of  a  boat.  (The  broad  blade,  approximately  fixed  in  the  water, 
acts  as  fulcrum.)  A  door,  when  used  to  crack  a  nut  in  the  hinge. 

Double  levers.     A  pair  of  bellows.     A  pair  of  nutcrackers. 

(3)  Regard  B  as  the  fulcrum,  but  P  as  the  weight  and  P  +  Q 
as  the  power.     The  lever  is  said  to  be  of  the  third  class,  and  there 
is  always  mechanical  disadvantage.     As  above, 

product  of  power  and  its  arm  =  product  of  weight  and  its  arm. 

Examples : — Single  levers.  Most  of  the  limbs  of  the  body  are 
of  this  class.  Thus  the  forearm 
moves  about  the  elbow -joint  as  a 
fulcrum.  The  ^power  is  applied 
(very  obliquely  too)  by  the  biceps 
muscle.  The  mechanical  disad- 
vantage is  very  great,  and  the 
muscles  must  possess  great 
strength ;  but  this  could  not  be 
avoided  unless  the  human  body  FifT  10 

were    constructed    so    as    to   re- 
semble an  animated  derrick,  which  would  be  awkward  for  loco- 
motion and  activity. 

Doable  levers.     A  pair  of  sugar  tongs.     A  pair  of  tweezers, 
c. 


18 


MECHANICS 


[CHAP. 


Fig.  10  a.  A  striking  illustration  of  Levers  of  the  Third  Class  is  found  in  the 
Hydraulic  Scale  of  the  Emery  Testing  Engine  in  the  Testing  Laboratory  of  McGill 
University,  Montreal.  The  pressure  is  conveyed  from  the  ram  of  the  engine  to  the 
drum  above  Pl  through  copper  pipes.  This  is  applied  at  the  knife-edge  P1  to  the 
lever  whose  fulcrum  wflt  The  force  is  reduced  by  a  series  of  levers  of  the  third 
class,  and  conveyed  to  the  central  weighing  lever,  and  the  deflection  of  the  latter 
is  magnified  by  the  upper  lever.  The  weights  are  applied  automatically  by  raising 
the  four  handles  to  the  left. 


WHEEL   AND   AXLE 


19 


The  Wheel 
and  Axle. 


14.  A  straight  lever,  working  on  a  fixed  fulcrum,  can  only 
raise  the  weight  to  a  height  above  the  fixed 
fulcrum  equal 'to  the  length  of  the  short  arm. 
This  difficulty  is  got  over  in  the  second  of  the  mechanical  powers, 
the  Wheel  and  Axle.  It  consists  of  a  wheel  of  large  radius  rigidly 
bolted  to  an  axle  of  smaller  radius.  The  weight  is  hung  from  a 
cord  coiled  on  the  axle.  The  power  is  applied  to  the  large  wheel 
by  pulling  on  a  cord  coiled  round  its  circumference,  or  by  a 
handle  projecting  from  its  rim,  as  in  the  familiar  device  for  raising 
Avater  from  wells. 


Fig.  12. 

It  is  obvious  that  the  condition  of  equilibrium  is  the  same  as 
for  the  lever 


In  fact  at  any  given  instant  the  radii  AC,  OB  form  a  straight 
lever.  But  as  each  radius  moves  out  of  position,  the  next  takes 
its  place.  The  wheel  and  axle  may  therefore  be  regarded  as  a 
continuous  lever. 

Another  form  of  the  wheel  and  axle  is  the  Capstan,  where  the 
power  is  applied  by  handspikes,  and  the  resemblance  to  a  lever  is 
still  more  obvious. 


15. 


The  Pulley. 


The  simple  Pulley  is  a  wheel  with  grooved  edge  round 
which  a  cord  is  passed  and  supported  at  one  end. 
The   power   is   applied   to   the  other   end.     The 
weight  is  hung  from  the  axle  of  the  pulley. 

2—2 


20 


MECHANICS 


CHAP. 


At  any  instant  the  diameter  ACB  may  be  regarded  as  a  lever 
with  fulcrum  at  A.     Hence 

PxAB=WxAC, 


__ 
W~AB~2' 


(Otherwise  thus  :  —  The  tension  of 
the  string  must  be  the  same  on  each 
side,  or  else  the  pulley  would  turn. 
Hence  the  weight  is  supported  by 
two  pulls  applied  at  A  and  B,  each 
equal  to  P.  Therefore  P=  W/2. 
This  is  clear  enough  if  each  string 
is  held  up  by  a  man.  It  makes 
no  difference,  however,  if  one  end, 
instead  of  being  held  by  a  man, 
is  fastened  to  a  fixed  support.) 

If  the  weight  of  the  pulfey  is 
too  great  to  be  neglected,  let  it  be  w. 


Then 


P  = 


As  the  pulley  rises,  fresh  diameters  take  the  place  of  AB,  and 
since  the  fulcrum  moves,  we  may  regard  the  pulley  as  a  Travelling 
lever. 

16.     A  single  moveable  pulley  only  enables  us  to  double  the 

systems  of         force  at  our  disposal.   By  combining  several  pulleys 

Pulleys.  we   may  increase   the   mechanical   advantage   to 

any  extent.     The  following  combinations  are  in  common  use,  or 

interesting  historically. 

(1)     Archimedes'  System.     (Fig.  14.) 

By  the  principle  of  the  simple  pulley  the  tension  in  each  string 
is  double  that  of  the  string  next  above  it.  The  weight  is  double 
the  tension  of  the  last  string.  Hence  if  there  be  n  moveable  pulleys, 

W  x  2"  =  P, 


and 


PULLEYS 


21 


(2)     The  Pulley  Block.     (Fig.  15.) 

There  are  two  blocks,  each  containing  several  pulley-wheels,  or 
sheaves,  on  the  same  axle.  The  string  is  fastened  to  one  of  the 
blocks,  and  then  carried  round  all  the  sheaves  as  in  the  figure. 


Fig.  14. 


Fig.  15. 


The  tension  of  the  string  is  the  same  throughout,  so  that  the 
weight  is  supported  by  as  many  tensions  each  equal  to  the  power 
as  there  are  strings  at  the  lower  block.  Count  these,  and  let  their 
number  be  n.  Then 


P_ 

W 


If  it  be  desired,  allowance  can  be  made  for  the  weights  of  the 
pulleys  as  before. 


EXAMPLES. 

1.  A  pump  handle  is  3  ft.  8  in.  long,  and  works  on  a  pivot  4  in.  from  the 
end  attached  to  the  pump  rod.  What  force  is  applied  to  the  pump  rod  when 
the  handle  is  pushed  down  with  a  force  of  10  Ibs.  weight  ? 


22  MECHANICS  [CHAP,  n 

2.  A  safety  valve  consists  of  a  circular  hole,  |-  inch  in  diameter,  closed  by 
a  plunger  attached  to  a  light  horizontal  hinged  bar  one  inch  from  the  hinge. 
A  weight  of  1  Ib.  slides  on  the  bar.     How  far  from  the  hinge  must  it  be  set  if 
the  steam  is  to  blow  off  at  160  Ibs.  on  the  square  inch  ? 

3.  An  oarsman  weighing  180  Ibs.  pulls  horizontally  at  the  handle  of  an  oar 
so  as  just  to  lift  his  weight  from  the  seat.     The  stretcher  against  which  his 
feet  press  is  16  inches  below  the  level  of  his  hands,  and  distant  2  ft.  from 
the  vertical  through  his  centre  of  gravity.     What  is  the  force  applied  to  the 
oar? 

4.  If,  in  example  3,  the  rowlock  is  at  one-quarter  of  the  distance  from  the 
hands  to  the  blade  of  the  oar  in  the  water,  what  propelling  force  could  eight 
such  oarsmen  apply  to  the  boat  ? 

5.  Six  men  work  a  capstan  using  handspikes  projecting  5  ft.  3  in.  from 
the  centre.     The  barrel  on  which  the  rope  is  coiled  is  2  ft.  3  in.  in  diameter. 
What  force  must  each  man  exert  in  order  to  raise  a  weight  of  a  ton  and  a  half  ? 

6.  The  rope  of  the  simple  pulley,  Fig.  13,  is  carried  over  a  fixed  pulley 
and  held  by  a  man  who  supports  himself  by  standing  in  the  hook  attached  to 
the  moveable  pulley.   What  is  the  pull  on  the  rope  if  the  man  weighs  180  Ibs.  ? 

7.  If  there  are  four  pulleys  in  the  system  of  Archimedes,  what  force  is 
required  to  support  a  weight  of  2  cwt.  (1)  when  the  weight  of  the  pulleys  is 
neglected,  (2)  when  each  pulley  weighs  8  Ibs.  ? 


CHAPTER  IIL 

THE  CENTRE   OF   GRAVITY. 

17.  THE  principle  of  the  lever  shews  us  that  two  weights 
rigidly  attached  to  a  light  rod  will  balance  if  their  moments  about 
the  fulcrum  are  equal  and  opposite,  and  that  the  fulcrum  must  be 
supported  by  a  force  equal  to  the  sum  of  the  weights.  This 
principle  may  be  generalised  in  two  ways. 

(1)  Let  another  pair  of  weights  be  attached  to  the  same  rod. 
Then  if  their  moments  about  the  fulcrum  are  equal  and  opposite, 
they  also  will  balance.  It  is  a  fact  of  experience  that  the  presence 
of  the  one  pair  in  no  way  interferes  with  the  equilibrium  of  the 
other.  The  same  is  true  for  any  number  of  pairs. 


A' 


P     \ 


Fig.  16. 

Hence  any  number  of  weights  at  different  distances  on  a  rod 
will  balance  provided  that  the  sum  of  the  moments  on  one  side 


24  MECHANICS  [CHAP. 

of  the  fulcrum  is  equal  to  the  sum  of  the  moments  on  the  other 
side. 

(2)  The  rod  may  be  turned  through  any  angle  about  the 
fulcrum,  and  yet  equilibrium  will  subsist. 

For  by  similar  triangles 

OA'     CA 


Q_CA_  CA' 
P~  GB~  CB" 

PxCB'=QxCA', 

and  the  moments  are  still  equal. 

A  rod  so  weighted  that  the  sum  of  the  moments  on  each  side 
of  the  fulcrum  is  the  same  may  be  said  to  be  statically  symmetrical 
about  the  fulcrum. 

It  is  clear  that  there  may  be  as  many  such  rods  as  we  choose, 
all  rigidly  joined  at  the  fulcrum,  and  yet  the  whole  system  will 
balance  in  any  position  about  it. 

Now  the  objects  with  which  we  deal  in  Mechanics  consist  of 
innumerable  small  parts,  or  particles,  rigidly  joined  together  and 
each  possessing  its  own  weight.  Mechanical  problems  will  be 
enormously  simplified  if  we  can  find  for  any  object  the  point 
about  which  it  is  statically  symmetrical.  For  if  this  be  supported 
by  a  force  equal  to  the  total  weight  of  the  object,  equilibrium 
will  subsist,  since  the  object  will  certainly  balance  about  this 
point.  For  many  purposes  we  need  no  longer  consider  the 
myriads  of  small  weights,  but  replace  them  by  a  single  weight  at 
this  point.  Hence  the  point  is  called  the  Centre  of  Gravity  of 
the  object. 

Definition.  The  Centre  of  Gravity  of  a  body  is  the  point  about 
which  it  will  balance  in  all  positions. 

Two  things  should  be  noted  : 

(1)  It  is  not  sufficient  that  there  should  be  statical 
symmetry  in  one  direction,  say  right  and  left.  A  vertical  rod 
will  remain  at  rest  however  the  weights  are  distributed  on  it, 
even  though  all  of  them  should  be  above  the  fulcrum.  For 
since  all  the  perpendiculars  from  the  fulcrum  on  the  vertical 
lines  of  action  of  the  weights  are  zero,  the  moments  on  each 


Ill] 


THE   CENTRE   OF   GRAVITY 


25 


side  of  the  fulcrum  are  zero,  and  there  is  statical  symmetry 
horizon tally,  but  not  vertically. 

But  if  the  rod  be  turned  ever  so  slightly  from  the  vertical, 
equilibrium  is  at  once  destroyed.  If  an  object  be  found  to  balance 
about  a  point  in  more  than  one  position,  then  it  will  balance  in 
all  positions,  and  the  point  of  support  must  be  the  centre  of 
gravity. 

(2)  There  cannot  be  two  centres  of  gravity  for  the  same  body, 
for  if  the  body  were  turned  so  that  the  line  joining  the  two  centres 
was  horizontal,  the  moments  to  the  left  and  right  could  not  be 
equal  for  both  points  at  the  same  time. 

18.  Experiment.     Find  experimentally  the  centre  of  gravity 
of  a  flat  board. 

Bore  two  small  holes  near  the  rim,  and  suspend  the  board  from 
a  knitting-needle  passed  through  one 
of  them  at  A. 

Hang  from  the  needle  a  plumb- 
line  which  has  been  rubbed  with 
chalk.  By  plucking  the  line  and 
letting  it  spring  back  a  chalk  line 
may  be  traced  on  the  board.  Repeat 
the  process  using  the  other  hole  B. 
The  intersection  of  the  traces  is  the 
centre  of  gravity. 

19.  The  centre  of  gravity  can 
be  found  by  inspection  whenever  we 
can  discern  a  point  about  which  the 
object  is  symmetrical  in  all  direc- 
tions.  This  was  the  method  adopted 
by  Archimedes  in  his  proof  of  the 
Principle  of  the  Lever.     (§§  2 — 3.) 

It  will  now  be  clear,  as  observed  at  the  time,  that  the  Principle  of 
the  Centre  of  Gravity  is  nothing  but  the  Principle  of  the  Lever 
in  its  most  general  form.  The  rest  of  his  treatise  is  devoted  to 
finding  the  centres  of  gravity  of  some  of  the  more  familiar 
geometrical  figures. 


26 


MECHANICS 


[CHAP. 


Thus,  the  C.G.  of  a  straight  line  is  its  middle  point,  for  it  may 
be  divided  into  pairs  of  particles  equidistant  from  the  centre  on 
opposite  sides. 

The  C.G.  of  a  circle  or  of  a  f~~  ~7 

sphere  is  its  centre.  /  / 

A  parallelogram  may  be  di-          G/— 
vided  into  strips  parallel  to  one        / 
side,  AB,  each  of  which  is  bi-     /. 
sected  by  EF  joining  the  middle     D 
points  of  AB,  CD. 

The  C.G.  therefore  lies  in  EF.  Similarly,  it  lies  in  GH.  There- 
fore it  is  K. 

20.     To  find  the  centre  of  gravity  of  a  triangle,  ABC. 

A 


Fig.  18. 


B  D  O 

Fig.  19. 

Bisect  the  base  in  D,  and  join  AD. 

Divide  the  triangle  into  small  strips,  such  as  bdc  parallel  to 
the  base   BDC.      Then    each   strip   is   bisected,   for    by   similar 

triangles 

bd      Ad       dc 


Ill]  THE    CENTRE    OF   GRAVITY  27 

But  BD  =  DC.     Therefore  bd  =  dc. 

Hence  the  c.G.  of  each  strip,  and  therefore  of  the  whole 
triangle,  lies  in  AD. 

Similarly,  it  lies  in  BE,  if  E  is  the  middle  point  of  AC. 

.'.  it  is  the  point  G. 

Join  DE.     Then  since  CB,  CA  are  bisected  in  D  and  E,  DE  is 
parallel  to  AB,  and  we  have  by  similar  triangles 
GD_DE_DC_l 
GA~AB~  BG~2' 

The  C.G.  of  the  triangle  is  therefore  on  the  line  joining  the 
middle  point  of  the  base  to  the  vertex,  at  one-third  of  its  length 
from  the  lower  end. 

21.  To  find  the  c.G.  of  any  number  of  weights  spaced  out 
along  a  straight  line. 


I 

I 

1 

It's 

•> 

M!  +  W2 

Fig.  20. 

f 

If  the  C.G.  were  supported  by  a  force  equal  to  the  sum  of  the 
weights,  the  whole  would  remain  at  rest  ;  and  this  would  not  be 
altered  if  the  rod  were  produced  and  fixed  at  any  point,  say  0. 

But  the  rod  would  turn  about  0  unless  the  moments  of  the 
separate  weights  about  0  were  equal  to  the  moment  of  the 
supporting  force  in  the  opposite  direction. 

/.  (w1  +  w2  +  ...)x  OG  =  iv1x  OA+w2xOB+... 


Wl  +  W2  +  .  .  . 

Rule.  Hence  the  distance  of  the  c.G.  to  the  right  of  the 
vertical  through  any  point  0  is  found  by  dividing  the  sum  of 
the  products  of  each  weight  by  its  distance  from  this  vertical 
by  the  sum  of  all  the  weights. 


28 


MECHANICS 


[CHAP. 


Very  often  an  object  has  a  line  of  symmetry,  and  consists  of 
portions  with  known  centres  of  gravity  spaced  out  along  this 
line.  It  is  then  easy  to  find  its  C.G.  by  the  above  rule. 

Since  the  object  may  be  turned  so  as  to  have  any  line  within 
it  vertical,  the  same  rule  will  give  the  distance  of  the  C.G.  from 
any  line  we  choose.  The  C.G.  of  a  flat  body  of  any  shape  may 
thus  be  fixed  by  finding  its  distance  from  two  chosen  lines  at 
right  angles  to  each  other. 

22.     One  point  of  support. 

If  an  object  is  supported  at  one  point,  it  will  be  in  equilibrium 

Equilibrium  so  l°ng  as  ^s  C.G.  is  in  the  vertical  line  through 

under  Gravity.       foe  point  of  support.     For  since  its  weight  then 

acts  through  the  point  of  support,  there  is  no  moment  tending  to 

turn  it  about  that  point.     But   the   nature  of  the  equilibrium 

differs  greatly  according  to  the  position  of  the  C.G. 


G'    \ 


Pendulum  in  Billiard  Cue  in 

Stable  Equilibrium.  Unstable  Equilibrium. 

Fig.  21. 


Ill]  THE   CENTRE    OF   GRAVITY  29 

(1)  Stable  Equilibrium.     If  the  c.G. 'is  below  the  point  of 
support,  and  the  body  be  accidentally  disturbed,  the  moment  of 
the  weight  tends  to  bring  it  back   to  its  position  of  rest,  and 
equilibrium  is  restored.     In  this  case  the  object  is  said  to  be  in 
Stable  Equilibrium. 

(2)  Unstable  Equilibrium.     If  the  C.G.  is  above  the  point  of 
support,  and  can  descend  when  the  object  is  disturbed,  the  moment 
of  the  weight  tends  to  turn  the  object  still  farther  from  the  position 
of  rest,  so  that  on  the  occurrence  of  any  accidental  disturbance 
equilibrium  is  destroyed.     This  is  called  Unstable  Equilibrium. 

(3)  Neutral  Equilibrium.     If  the  C.G.  is  at  the  fixed  point  of 
support,  the  object  will  rest  in  any  position.     Such  equilibrium 
is  called  Neutral  Equilibrium.     It   occurs  also  when  the  point 
of  support  is  moveable  so  that  the  C.G.  remains  always  at  the 
same  height  above  it,  as  when  a  sphere  or  cylinder  rests  upon  a 
horizontal  plane. 

23.  Objects  standing  upon  a  base  will  be  in  equilibrium  so 

long  as  the  vertical  through  the  C.G.  falls  within 

Extended  Base. 

the  contour  of  the  base.  (If  the  base  has  pro- 
jecting points  and  retreating  bays,  the  contour  is  to  be  drawn  from 
point  to  point,  and  not  to  follow  the  inner  curve  of  the  ., 

bays.)  For  then  the  upward  pressures  from  the  base 
can  arrange  themselves  so  as  to  meet  and  balance  the 
weight  acting  vertically  through  the  C.G. 

The  gesticulations  of  a  person  walking  along  a 
narrow  plank  are  instinctive  efforts  to  bring  back  (by 
shooting  out  an  arm  or  a  leg)  the  C.G.  of  the  body  to 
the  vertical  over  the  line  joining  the  feet.  The  tight- 
rope dancer  aids  himself  by  a  balancing  pole  heavily 
weighted  at  each  end.  A  slight  motion  of  the  pole  and 
weights  suffices  to  move  the  C.G.  as  much  as  a  violent 
movement  of  the  limb,  and  thus  awkward  and  inelegant 
gyrations  are  avoided. 

24.  In    bicycle    riding    the    greater    part    of    the  B'-l 
balancing    depends    on    this    principle,    though    some 
help    for    steadiness    is    derived    from    two    dynamical  Qi 
considerations   to   be   mentioned   later   (§8  70,  77).  Fig.  22. 


30  MECHANICS  [CHAP. 

Let  AB  be  the  ground  contacts  of  the  front  and  back  wheels, 
seen  from  above ;  G  the  C.G.  of  machine  and  rider.  If  G  moves  off 
the  base  line  AB,  say  to  the  right,  the  rider  at  once  feels  that  he 
is  falling  over  on  that  side.  By  turning  the  front  wheel  towards 
the  side  on  which  he  is  falling,  he  brings  A  to  A',  while  B  follows 
along  BA  to  B'}  and  the  base  is  again  beneath  the  C.G.  Hence  the 
rule,  so  contrary  to  the  beginner's  instincts,  that  the  wheel  must  be 
turned  towards  the  side  on  which  the  rider  is  falling. 


EXAMPLES. 

1.  Shew  that  the  centre  of  gravity  of  a  triangle  is  the  same  as  that  of 
three  equal  weights  placed  at  its  corners. 

2.  From  a  body  of  weight  W  and  centre  of  gravity  G  a  portion  is  cut 
away  whose  weight  is  W  and  centre  of  gravity  G'.     Shew  that  the  centre  of 

W 

gravity  of  the  remainder  is  G"tonG'G  produced,  where  G"G=  ™ — ™  G  'G. 

3.  Weights  of  2,  4,  6,  8,  10,  12  Ibs.  are  spaced  out  along  a  straight  line  at 
equal  distances  of  one  foot.     Find  their  centre  of  gravity. 

4.  A  figure  is  formed  of  a  square  of  side  a  and  an  isosceles  triangle 
described  on  one  of  the  sides  as  base.     Find  the  altitude  of  the  triangle  in 
order  that  the  figure  may  balance  about  that  side. 

5.  A  sphere  of  6  inches  radius  has  a  hollow  spherical  cavity  of  2  inches 
radius,  midway  between  the  centre  and  the  surface.     Find  the  distance  of  the 
C.G.  from  the  centre. 

6.  Where  must  a  circular  hole  2  inches  in  diameter  be  punched  out  of  a 
circular  plate  5  inches  in  diameter  in  order  that  the  distance  of  the  C.G. 
from  the  centre  may  be  half  an  inch  ? 

7.  The  mass  of  the  moon  is  '01137  times  that  of  the  earth.     Taking  the 
earth's  radius  at  3963  miles  and  the  distance  of  the  moon  from  the  earth's 
centre  at  60'27  radii  of  the  earth,  find  the  c.G.  of  the  earth  and  moon. 

8.  Shew  that  the  C.G.  of  a  pyramid  on  a  triangular  base  is  in  the  line 
joining  the  vertex  to  the  c.G.   of    the  base  at  one-quarter  of  its  length 
from  the  C.G.  of  the  base.     (Consider  slices  parallel  to  the  base  and  proceed 
as  in  §  20.) 


Ill]  THE    CENTRE   OF   GRAVITY  31 

9.  A  cylindrical  tin  can  (without  lid)  8  inches  in  diameter  and  one  foot 
high,  is  half  filled  with  water.     Find  the  c.G.  of  the  can  and  the  water,  if 
the  weight  of  the  can  is  one-quarter  of  that  of  the  water. 

10.  A  rod  balances  about  a  point  one-quarter  of  its  length  from  one  end. 
If  a  weight  of  2  Ibs.  is  attached  to  the  thin  end,  the  balancing  point  is  shifted 
8  inches  towards  that  end  ;  whereas  8  Ibs.  must  be  attached  to  the  thick  end 
to  shift  it  the  same  amount  in  the  other  direction.     Find  the  weight  and 
length  of  the  rod. 


CHAPTER  IV. 

THE  BALANCE. 

25.  ONE  of  the  most  important  cases  of  the  Lever  is  the 
Balance. 

In  principle  it  is  only  a  lever  with  equal  arms.  If  two  weights 
placed  at  their  extremities  balance  each  other,  they  must  be  equal. 


Fig.  23, 

Let  AB  be  the  beam  of  the  balance,  supported  at  its  middle 
point  C.  Let  AC  =  CB  =  l\  let  P  and  Q  be  the  weights  to  be 
compared ;  W  the  weight  of  the  beam  itself,  which  will  act  at  its 
centre  of  gravity  G. 

We.  shall  suppose  that  G  is  at  a  distance  h  below  C  when  the 
beam  is  horizontal.  Consider  what  would  happen  if  G  were 
(1)  above  C,  (2)  exactly  at  C.  (§  22.) 


CHAP.  IV]  THE   BALANCE  33 

If  the  beam  comes  to  rest  at  an  angle  6  with  the  horizon,  we 
have  by  the  principle  of  the  lever 

P  x  AD  =  Q  x  BE+  W  x  GF, 
P  Jcos0  =     Jcos<9+ 


If  the  weights  are  equal,  6  —  0,  and  the  beam  is  horizontal. 

26.  A  balance  is  sai(J  to  have  great  sensitiveness  when  a  very 
small  difference  of  weights  causes  a  great  deflection.    To  construct 
a  sensitive  balance,  we  must  make  I  large,  and  W  and  h  small,  i.e. 

(1)  the  beam  must  be  long; 

(2)  the  beam  must  be  light  ; 

(3)  the  centre  of  gravity  of  the  beam  must  be  very  near  (but 
not  at)  the  point  of  support. 

Besides  these  three  requisites  the  mechanician  has  also  to 
arrange, 

(4)  that  C  shall  be  exactly  on  the  line  AB  (for  if  it  is  above 
or  below,  the  sensitiveness  will  be  different  for  different  loads. 
Ex.  8); 

(5)  that  there  shall  be  as  little  friction  as  possible  at  the 
points  where  the  beam  and  the  weights  are  supported  (for  friction 
would  hinder  the  free  turning  of  the  beam,  and  perhaps  cause  the 
weights  not  to  hang  exactly  in  the  vertical  through  the  points  of 
support)  ; 

(6)  that  the  time  of  swing  shall  not  be  too  great. 

The  conditions  for  this  last  requisite  will  be  understood  later 
(§  270),  but  they  cannot  be  satisfied  consistently  with  (1)  and  (3). 
Hence  a  compromise  must  be  effected.  Where  accuracy  is  all- 
important  (6)  must  be  given  up,  and  weighing  will  occupy  much 
time.  For  rapid,  but  rough,  weighing  (1)  and  (3)  are  sacrificed. 

27.  Fig.  24  represents  a  16-inch  Oertling  Balance. 

The  metal  beam  is  constructed  like  a  girder  so  as  to  combine 
lightness  with  great  rigidity.  The  point  of  support  is  a  knife- 
edge  of  polished  agate  projecting  downwards  at  the  centre  and 
resting  on  an  agate  plane.  Two  other  agate  knife-edges  project 
upwards  at  the  ends  of  the  beam,  and  on  these  rest  agate  planes 

c.  3 


MECHANICS 


[CHAP. 


from  which  the  scale-pans  are  hung.  Above  the  centre  of  the 
beam  may  be  seen  the  gravity  bob,  a  small  brass  weight  which 
moves  up  and  down  on  a  fine  screw.  By  means  of  it  the  centre  of 
gravity  may  be  raised  or  lowered  very  gradually,  and  can  be 
adjusted  so  as  to  be  sometimes  not  more  than  one  thousandth  of 
an  inch  below  the  knife-edge  supporting  the  beam.  A  small  vane, 
or  flag,  is  also  seen,  which  may  be  turned  to  the  right  or  left  so  as 
to  correct  slight  deviations  from  the  horizontal  when  the  beam  is 
unloaded. 


Fig.  24. 

A  long  pointer,  attached  at  right  angles  to  the  centre  of  the 
beam,  moves  over  the  divisions  of  an  ivory  scale  at  the  foot  of  the 
pillar,  and  should  point  to  zero  when  the  beam  is  horizontal. 

To  protect  the  agate  edges  from  injury  while  the  weights  are 


IV]  THE   BALANCE  35 

being  changed,  and  from  unnecessary  wear  and  tear,  a  supporting 
framework  can  be  raised  (by  turning  the  knob  A)  so  as  to  gently 
lift  the  beam  off  the  agate  plane  on  which  it  rests,  and  the  bars 
supporting  the  scale-pans  off  their  knife-edges. 

The  beam  between  the  central  knife-edge  and  one  or  both  of 
those  at  its  ends  is  graduated  into  ten  equal  parts,  and  each  of 
these  has  again  ten  divisions.  By  a  lever  worked  from  outside  the 
case  a  small  "  rider "  (fl )  of  platinum  wire,  weighing  one  centi- 
gram, can  be  placed  on  any  division  of  the  beam.  The  case 
protects  the  balance  from  disturbance  by  air  currents  during  the 
final  stages  of  the  weighing.  It  rests  upon  four  levelling  screws, 
and  is  provided  with  two  spirit  levels  at  right  angles  to  each  other. 

28.     With  a  properly  constructed  and  well  adjusted  balance 
use  of  the  it   is   for   most   purposes   enough   to   proceed   as 

Balance.  follows. 

Lower  the  supports  by  gently  turning  the  knob  till  the  beam 
is  free,  and  see  whether  the  pointer  rests  at  zero,  or  swings  to 
equal  distances  on  each  side  of  it,  when  the  pans  are  empty. 

Raise  the  supports  again,  and  place  the  object  to  be  weighed 
in  one  scale-pan,  and  weights  in  the  other.  A  slight  turn  of  the 
knob  will  shew  which  way  the  pointer  begins  to  move,  and  after 
again  supporting  the  beam,  weights  must  be  added  or  subtracted 
till,  when  the  beam  is  set  free,  the  pointer  swings  slowly  back- 
wards and  forwards  within  the  limits  of  the  scale.  The  case  may 
then  be  closed,  and  the  "rider"  placed  on  the  beam,  and  shifted 
till  the  pointer  rests  at  zero,  or  swings  equally  on  both  sides  of  it. 
The  beam  must  be  brought  to  rest  on  the  supports  every  time  the 
rider  is  to  be  moved. 

The  weights  in  the  scale-pan  are  then  recorded  before  any  of 
them  are  removed,  and  allowance  made  for  the  rider  as  follows. 

The  rider  weighs  10  milligrams.  If  it  is  placed  over  the 
division  marked  7,  for  instance,  it  will  balance  a  weight  of 
7  milligrams  placed  in  the  other  scale,  (which  hangs  at  division  10 
on  the  other  arm),  by  the  principle  of  the  lever,  for 

10x7  =  7  x  10. 

Since  each  division  of  the  beam  is  subdivided  into  tenths  and 

3—2 


36  MECHANICS  [CHAP. 

we  can  estimate  tenths  of  one  of  the  subdivisions,  we  have  the 
means  of  reading  to  hundredths  of  a  milligram. 

29.  But  if  such  accuracy  as  this  is  desired,  it  is  better  to 
proceed  by  what  is  known  as  the  Method  of  Oscillations.  We  do 
not  wait  for  the  beam  to  come  to  rest,  but  calculate  the  point  at 
which  it  will  stop  in  course  of  time  by  observing  the  oscillations  of 
the  pointer  on  the  scale. 

Suppose  the  scale  has  10  divisions  on  each  side  of  the  zero,  and 
we  read  the  turning  points  to  tenths  of  a  division  for  three 
successive  swings  thus: 

Left  Eight 

7-3 

6-2 

7-1 

The  mean  of  the  two  readings  on  the  right  is  7*2 ;  and  the 

—  6'2  +  7*2 

mean  of  this  with   the  reading   on   the   left   is =  *5. 

2i 

This  is  where  the  pointer  would  come  to  rest  if  left  to  itself  for  a 
quarter  of  an  hour. 

It  would  not  be  right  to  take  the  mean  of  one  observation  on 
each  side.  For  the  swings  are  gradually  decreasing,  and  we  ought 
to  compare  with  the  single  swing  on  the  left,  —  6' 2,  such  a  swing 
to  the  right  as  would  have  been  made  at  the  same  stage  of  their 
decay.  The  vibrations  decay  very  regularly,  so  that  the  mean 
between  the  two  swings  on  the  right  will  represent  what  a  right- 
hand  swing  would  have  been,  had  it  been  made  at  the  moment 
when  the  pointer  was  actually  swinging  6 '2  divisions  to  the  left. 
We  may  with  advantage  observe  more  than  three  swings,  but 
there  must  always  be  one  more  swing  on  one  side  than  on  the 
other. 

Using  this  method  we  determine : 

(1)  the  position  of  rest  with  the  pans  empty.     This  may  be 
called  the  true  zero  reading ; 

(2)  the  reading  when  weights  and  rider  are  adjusted  to  the 
nearest  whole  milligram  less  than  the  object ; 

(3)  the  reading  when  the  rider  is  shifted   to   increase   the 
weight  by  one  milligram. 


IVJ  THE   BALANCE  37 

Thus  let  the  readings  be: 

(1)  Pans  unloaded  (true  zero)         +    '5, 

(2)  With  object  and  weights  36*324  gms.     ...     -  4'2, 

(3)  „  „  36-325  gms.      ...     +8'6. 

The  addition  of  1  mgm.  makes  a  difference  of  12'8  divisions. 
This  measures  the  sensitiveness  of  the  balance  for  the  given  load. 

To  bring  the  pointer  from  position  (2)  to  the  true  zero  a 
difference  of  4*7  divisions  must  be  effected,  and  hence  a  weight  of 
4'7/12'8  mgms.,  i.e.  "37  mgms.,  must  be  added. 

The  true  weight  is  therefore  36'32437  gms. 

30.  Such  perfection  has  been  attained  in  the  construction  of 
balances  that  a  difference  of  one  milligram  may  be  detected  in  a 
load  of  one  kilogram,  i.e.  one  part  in  a  million.     In  a  first-class 
balance  the  arms  may  be  so  nearly  equal  in  length  as  not  to  differ 
by  one  part  in  50,000 ;  the  knife-edges  so  keen  as  to  be  less  than 
one  two-hundred  thousandth  of  an  inch  wide ;  and  the  centre  of 
gravity  may  be  less  than  one-thousandth  of  an  inch  below  the 
point   of    support.     Such    instruments    demand    great    care    in 
handling,  and  the  following  precautions  should  be  strictly  observed 
by  those  who  use  them. 

(1)     No  one  should  alter  any  of  the  adjustments  except  those 
responsible  for  the  care  of  the  instrument. 

Precautions. 

(2)  No  change  in  the  object,  weights,  or 
position  of  the  rider  should  be  made,  nor  must  the  scale-pans  or 
any  part  of  the  swinging  system  be  touched  except  after  the  beam 
has  been  arrested  by  turning  the  milled  knob. 

(3)  The  knob  should  be  turned  gently,  and  so  as  to  arrest 
the  beam  as  nearly  as  possible  at  the  middle  of  the  swing.     The 
great  object  is  to  avoid  the  smallest  jerk  or  jar,  as  these  are  likely 
to  injure  the  agate  knife-edges. 

(4)  The  weights  must  not  be  touched  except  with  the  pliers 
provided  for  the  purpose. 

31.  The  chief  source  of  error  in  an  accurate  weighing,  so  far 
as  the  balance  itself  is  concerned,  is  a  slight  difference  in  the 
lengths  of  the  arms.     This  error  may  be  avoided  in  two  ways. 


38 


MECHANICS 


[CHAP. 


(1)  Borda's  method  consists  in  counterpoising  the  object  by 
weights,  small  shot,  fine  sand,  or  thin  paper;  and  then  substituting 
standard  weights  for  the    object   till   they  exactly  balance   the 
counterpoise.     It  is  clear  that  the  arms  need  not  be  equal  in  this 
method. 

(2)  Gauss  devised  the  method  of  weighing  the  object  in  each 
scale  successively. 

Let  a,  b  be  the  lengths  of  the  arms ;    W1}  W2  the  apparent 


Fig.  25. 


OF 

FO 
IV]  THE  BALANCE  39 

weights  when  the  object  hangs  at  a  and  b  respectively;   W  the 
real  weight.     Then  by  the  principle  of  the  lever 

W  x  a  =  Wl  x  b 


and  W  =JW1.  W,. 

Since  TTj  and  TF2  are  very  nearly  equal,  it  is  generally  accurate 
enough  to  take  the  arithmetical  mean  instead  of  the  geometric 
mean. 

32.     As  an  instance  of  a  balance  for  rough  but  rapid  weighing 
we  may  take  the  common  steelyard,  to  be  found 

The  steelyard.  * 

at  any  railway  station. 

A  platform  is  hung  from  the  short  end  of  a  balance,  very 
near  the  fulcrum.  Heavy  weights  can  be  hung  on  a  hook  at 
the  end  of  the  long  graduated  arm,  and  a  small  rider  slides  on  the 
latter,  as  in  the  fine  balance  just  described.  It  is  evident  that  the 
divisions  can  be  adjusted  so  as  to  read  in  any  units  that  may  be 
desired. 


EXAMPLES. 

1.  A  balance  is  horizontal  when  unloaded.   But  an  object  weighs  20'4  gms. 
in  one  scale  and  20'8  gms.  in  the  other.    What  is  the  matter  with  the  balance 
and  what  is  the  true  weight  ? 

2.  An  object  weighs  12  and  14  gms.  respectively  in  the  two  scales  of 
a  balance.     What  is  the  error  if  its  weight  is  taken  as  13  gms.  ? 

3.  The  pans  of  a  balance  are  not  quite  of  the  same  weight,  but  the  arms 
are  equal.     Shew  that  the  weight  of  an  object  which  appears  to  weigh    W^ 

"W  -4-  "W 
and  W2  in  the  two  scales  is  — ^ — -  ;  and  that  the  difference  of  the  weights 

of  the  pans  is    -^ — - . 

4.  A  body  whose  weight  is  12  Ibs.  appears,  in  one  scale  of  a  balance,  to 
weigh  12  Ibs.  6  oz.     Find  its  apparent  weight  in  the  other  scale. 


40  MECHANICS  [CHAP.  IV 

5.  One  arm  of  a  balance  is  9  inches  long  and  the  other  10  inches.     Shew 
that  if  the  seller  puts  the  substance  to  be  weighed  as  often  in  one  scale  as  in 
the  other,  he  loses  f  °/0  on  his  transactions. 

6.  In  a  balance  with  unequal  arms  P  appears  to  weigh  Q,  and  Q  appears 
to  weigh  R  ;  what  does  R  appear  to  weigh  ? 

7.  The  beam  of  a  balance  is  18  inches  long,  and  an  object  appears  to 
weigh  20-34  gins,  in  one  pan,  and  20*87  gms.  in  the  other.     How  much  must 
the  fulcrum  be  shifted  to  make  the  balance  true  ? 

8.  In  a  balance  the  distance  between  the  knife-edges  supporting  the 
scale-pans  is  21.     The  central  knife-edge  is  at  a  perpendicular  distance  x 
above  the  middle  point  of  this  line,  and  the  centre  of  gravity  is  distant 
h  below  the  central  knife-edge.     If  weights  w19  w2  are  placed  in  the  pans,  and 
w  be  the  weight  of  the  balance,  shew  that  the  beam  will  come  to  rest  at  an 
angle  6  with  the  horizon,  where 


.  h 
Hence  shew  that  the  sensitiveness  decreases  as  the  loads  increase. 


CHAPTER  V. 

STEVINUS  OF  BRUGES— THE  PRINCIPLE   OF  THE 
INCLINED   PLANE. 

"Wonder  en  is  gheen  wonder." 

33.  THREE  of  the  Mechanical  Powers  remain  to  be  considered, 
viz.,  the  Inclined  Plane;  the  Wedge,  or  Double  Inclined  Plane; 
and  the  Screw,  which  consists  of  an  inclined  plane  wrapped  round 
a  cylinder,  and  may  be  regarded  as  a  Travelling  Inclined  Plane. 

From  the  time  of  Archimedes  nothing  of  importance  was 
effected  in  Mechanical  theory  for  nearly  two  thousand  years, 
when  Simon  Stevin  of  Bruges  (1548 — 1620)  established  the 
principle  of  the  Inclined  Plane.  His  discovery  constitutes  the 
second  step  in  the  historical  development  of  Mechanics.  Its 
importance,  and  the  beautiful  ingenuity  of  the  proof,  make  it 
worth  while  to  study  the  proof  in  his  own  words. 

Stevin  not  only  built  upon  this  foundation  the  theory  of 
pulleys  and  the  lever,  and  many  propositions  of  modern  Mechanics, 
but  applied  his  knowledge  to  practical  questions  such  as,  for 
instance,  the  design  of  the  machines  by  which  the  Dutch  fisher- 
men hauled  their  boats  above  high-water  mark;  the  best  form 
of  bit  for  the  management  of  horses  (at  the  request  of  Maurice 
of  Nassau,  Prince  of  Orange) ;  and  the  art  of  fortification.  Readers 
of  Tristram  Shandy  will  remember  that  his  work  was  Uncle  Toby's 
constant  companion. 

34.  The  problem  to  be  solved  was  this. 

A  body  resting  upon  a  horizontal  plane  requires  no  force  to 


MECHANICS 


[CHAP. 


support  it.  Let  it  be  attached  by  a  string  to  some  point  in  the 
plane,  and  let  the  plane  be  tilted 
till  it  becomes  vertical,  so  that 
the  body  hangs  freely  by  the 
string.  The  tension  of  the  string 
must  now  be  equal  to  the  full 
weight  of  the  body.  In  the  inter- 
mediate positions  the  tension  will 
be  something  between  the  weight 
of  the  body  and  zero.  What  is 
the  law  connecting  the  tension 
and  the  weight  for  any  given 
slope  of  the  plane  ?  That  is  the 
principle  to  be  discovered. 

35.  Here  is  Stevin's  solution,  arrayed  in  all  the  elaborate 
stevin's  Prin-  forms  of  a  proposition  of  Euclid.  (Elements  of 
ciple-  Statics,  Book  I.  Proposition  xix.) 

If  a  triangle  has  its  plane  perpendicular  to  the  horizon,  and 
its  base  parallel  to  it;  and  on  each  of  the  two  other  sides  a 
spherical  mass  of  equal  weight ;  the  power  of  the  left-hand 
weight  shall  be  to  the  power  of  the  right-hand  weight  as  the 
right  side  is  to  the  left  side. 


Fig.  26. 


M 


Fig.  27.     (From  Stevinus.) 


V]  STEVINUS.      THE   INCLINED   PLANE  43 

Given.  Let  ABC  be  a  triangle,  having  its  plane  perpendicular 
to  the  horizon,  and  its  base  AC  parallel  to  the  horizon ,  and  let 
there  be  on  the  side  AB  (which  is  double  of  BC)  a  globe  D,  and 
on  BC  another  E,  equal  in  weight  and  magnitude. 

Required.  To  prove  that  as  the  side  AB  (2)  is  to  the  side 
BC  (1),  so  is  the  power  of  the  weight  E  to  that  of  D. 

Preparation.  Let  there  be  fitted  round  the  triangle  a  circuit 
of  fourteen  globes,  equal  in  weight  and  size,  and  equidistant,  as 

D,  E,  F,  G,  Hy  /,  K,  L,  M,  N,  0,  P,  Q,  R,  threaded  on  a  cord  passing 
through  their  centres,  so  that  they  can  turn  on  the  said  centres, 
and  that  there  may  be  two  globes  on  the  side  BCy  and  four  on  AB; 
and  thus  as  line  is  to  line,  so  the  number  of  globes  to  the  number 
of  globes ;  let  there  be  three  fixed  points  at  S,  Ty  F,  over  which 
the  cord  can  slip,  and  let  the  two  parts  above  the  triangle  be 
parallel  to  its  sides  AB,  BC\  so  that  the  whole  can  turn  freely  and 
without  hindrance  on  the  said  sides  AB,  BC. 

If  the  power  of  the  weights  D,  R,  Q,  P,  be  not  equal  to  the 
power  of  the  two  globes  E,  F,  the  one  side  will  be 

Demonstration .  . 

more  powerful  than  the  other.  If  it  be  possible, 
let  the  four  D,  R,  Q,  P,  be  more  powerful  than  the  two  E,  F\  but 
the  four  0,  N,  M,  L,  are  equal  to  the  four  G,  H,  I,  K ;  wherefore 
the  side  of  the  eight  globes  D,  R,  Q,  P,  0,  N,  M,  L,  will  be  more 
powerful  in  consequence  of  their  arrangement  than  the  six, 

E,  F,  G,  H,  I,  K,  and   since   the   heavier   side   overcomes   the 
lighter,   the   eight   globes  will   descend   and   the   other  six  will 
rise. 

Let  it  be  so,  and  let  D  arrive  where  0  is  at  present,  and  so  for 
all  the  others;  viz.  let  E,  F,  G,  H,  arrive  where  P,  Q,  R,  D  are  now, 
and  7,  K,  where  E,  F  are.  Nevertheless  the  circuit  of  globes  will 
have  the  same  configuration  as  before,  and  for  the  same  reason  the 
eight  globes  will  have  the  advantage  in  weight  and  in  falling  will 
cause  eight  others  to  come  into  their  places,  and  so  this  movement 
will  have  no  end,  which  is  absurd. 

The  proof  will  be  the  same  for  the  other  side. 

Therefore  the  part  of  the  circuit,  D,  R,  Q,  P,  0,  N,  M,  L,  will 
be  in  equilibrium  with  the  part,  E,  F,  G,  H,  /,  K. 

Take  away  from  the  two  sides  the  weights  which  are  equal  and 


44  MECHANICS  [CHAP. 

similarly  situated,  as  are  the  four  globes,  0,  Nt  M,  L,  on  the  one 
part,  and  the  four,  G,  Ht  I,  Ky  on  the  other  part. 

The  remaining  four,  D,  R,  Q,  P,  will  be,  and  will  remain,  in 
equilibrium  with  the  two,  E,  F. 

Wherefore  E  will  have  double  the  power  of  D. 

As  the  side  BA  (2)  is  to  the  side  EG  (1),  so  is  the  power  of  E 
to  the  power  of  D. 


A  c  A  c 

Fig.  28.  Fig.  29. 

Corollary  1.  Let  ABC  be  a  triangle  as  before,  and  AB 
double  of  BC;  and  let  D,  a  globe  on  AB,  be  double  of  E  on  EC. 
It  appears  that  D,  E  will  be  in  equilibrium. 

Wherefore  as  A  B  is  to  BC,  so  is  the  globe  D  to  the  globe  E. 

Corollary  2.  Let  now  one  of  the  sides  of  the  triangle,  as  BC 
(which  is  half  of  AB),  be  perpendicular  to  AC.  The  globe  D, 
which  is  double  of  E,  will  still  be  in  equilibrium  with  E.  For  as 
the  side  A  B  is  to  BC,  so  is  the  globe  D  to  the  globe  E. 

In  the  last  corollary  the  tension  of  the  string  supporting  D  on 
the  inclined  plane  is  evidently  equal  to  the  weight  of  E  hanging 
freely. 

Thus  the  principle  is  reached  that  the  force  required  to  support 
a  body  resting  on  an  inclined  plane  is  to  the  weight  of  the  body  as 
the  height  of  the  plane  is  to  its  length  (along  the  slope). 

36.     Before  going  farther  the  student  should  immediately  test 
Experimental        this  important  principle  for  himself, 
verification.  Experiment.     With  the  apparatus  figured,  or 

some  simpler  arrangement,  make  a  series  of  observations  with 


STEVINUS.      THE   INCLINED   PLANE 


45 


different  slopes  and  different  weights  and  verify  that  in  every 

case 

P  _  Height  of  Plane 

W "Length  of  Plane 
where  a  is  the  inclination  of  the  inclined  plane  to  the  horizontal. 


sin  a, 


Fig.  30. 


37. 


In  all  probability  the  student,  especially  if  familiar  with 
criticism  of          mathematical  demonstrations,  will  feel  a  curious 
stevin's  Proof.       sudden  enlightenment,  and  intensity  of  conviction 
on  first  grasping  the  point  of  Stevin's  proof, — far  more  than  he 
will  derive  from  the  results  of  his  direct  experiments,  which  will 
give  only  approximate  values  owing  to  the  effects  of  friction  and 
the  difficulty  of  all  exact  measurements.     How  is  this  ?     Do  the 
truths  of  science  rest  after  all  on  a  priori  reasoning,  rather  than 


46  MECHANICS  [CHAP. 

on  observation  and  experience?  The  following  remarks  of  Professor 
Mach  on  this  point  are  so  instructive  that  I  venture  to  quote  them 
in  full ;  they  should  be  carefully  studied. 

"Unquestionably  in  the  assumption  from  which  Stevinus 
starts,  that  the  endless  chain  does  not  move,  there  is  contained 
primarily  only  a  purely  instinctive  cognition.  He  feels  at  once, 
and  we  with  him,  that  we  have  never  observed  anything  like  a 
motion  of  the  kind  referred  to,  that  a  thing  of  such  a  character 
does  not  exist.  This  conviction  has  so  much  logical  cogency  that 
we  accept  the  conclusion  drawn  from  it  respecting  the  law  of 
equilibrium  on  the  inclined  plane  without  the  thought  of  an 
objection,  although  the  law  if  presented  as  the  simple  result  of 
experiment,  or  otherwise  put,  would  appear  dubious.  We  cannot 
be  surprised  at  this  when  we  reflect  that  all  results  of  experiment 
are  obscured  by  adventitious  circumstances  (as  friction,  &c.),  and 
that  every  conjecture  as  to  the  conditions  which  are  determinative 
in  a  given  case  is  liable  to  error.  That  Stevinus  ascribes  to 
instinctive  knowledge  of  this  sort  a  higher  authority  than  to 
simple,  manifest,  direct  observation  might  excite  in  us  astonishment 
if  we  did  not  ourselves  possess  the  same  inclination.  The  question 
accordingly  forces  itself  upon  us:  Whence  does  this  higher 
authority  come  ?  If  we  remember  that  scientific  demonstration, 
and  scientific  criticism  generally,  can  only  have  sprung  from  the 
consciousness  of  the  individual  fallibility  of  investigators,  the 
explanation  is  not  far  to  seek.  We  feel  clearly  that  we  ourselves 
have  contributed  nothing  to  the  creation  of  instinctive  knowledge, 
that  we  have  added  to  it  nothing  arbitrarily,  but  that  it  exists 
in  absolute  independence  of. our  participation.  Our  mistrust  of 
our  own  subjective  interpretation  of  the  facts  observed,  is  thus 
dissipated." 

"  Stevinus'  deduction  is  one  of  the  rarest  fossil  indications  that 
we  possess  in  the  primitive  history  of  Mechanics,  and  throws  a 
wonderful  light  on  the  process  of  the  formation  of  science 
generally,  on  its  rise  from  instinctive  knowledge.  We  will  recall 
to  mind  that  Archimedes  pursued  exactly  the  same  tendency  as 
Stevinus,  only  with  much  less  good  fortune.  In  later  times,  also, 
instinctive  knowledge  is  very  frequently  taken  as  the  starting  point 
of  investigations.  Every  experimenter  can  daily  observe  in  himself 


y]  STEVINUS.      THE  INCLINED   PLANE  47 

the  guidance  that  instinctive  knowledge  furnishes  him.  If  he 
succeed  in  abstractly  formulating  what  is  contained  in  it,  he  will 
as  a  rule  have  made  an  important  advance  in  science." 

"  Stevinus'  procedure  is  no  error.  If  an  error  were  contained  in 
it,  we  should  all  share  it.  Indeed,  it  is  perfectly  certain  that  the 
union  of  the  strongest  instinct  with  the  greatest  power  of  abstract 
formulation  alone  constitutes  the  great  natural  enquirer." 

(Mach,  The  Science  of  Mechanics.  Translation  by  Thomas  J.  McCormack, 
p.  26.) 

And  again:  "The  reasoning  of  Stevinus  impresses  us  as  so 
highly  ingenious,  because  the  result  at  which  he  arrives  apparently 
contains  more  than  the  assumption  from  which  he  starts.  If 
Stevinus  had  distinctly  set  forth  the  entire  fact  in  all  its  aspects, 
as  Galileo  subsequently  did,  his  reasoning  would  no  longer  strike 
us  as  ingenious;  but  we  should  have  obtained  a  much  more 
satisfactory  and  clear  insight  into  the  matter." 

We  shall  see  later  how  Galileo  regarded  this  principle. 


EXAMPLES. 

1.  A  train  of  200  tons  weight  rests  on  an  incline  of  1  in  80.     There  is  a 
resistance  to  motion  due  to  friction  equivalent  to  an  opposing  force  of  16  Ibs. 
per  ton  weight.    What  force  is  required  (1)  to  prevent  the  train  from  running 
down  hill,  (2)  just  to  set  it  in  motion  up  hill  ? 

2.  To  pull  a  waggon  up  a  certain  hill  the  horse  has  to  exert  a  force  equal 
to  the  weight  of  480  Ibs.,  one  quarter  of  his  effort  being  employed  in  over- 
coming friction.     If  he  zig-zags  so  as  to  increase  the  distance  travelled  by 
one-third,  what  force  must  he  exert,  supposing  friction  to  be  the  same  as 
before  ? 

3.  Two  inclined  planes  of  equal  altitude  4  feet,  but  bases  3  feet  and  5  feet 
respectively,  are  placed  back  to  back,  and  two  weights  connected  by  a  smooth 
string  are  balanced  across  the  top,  one  on  each  incline.   Compare  the  weights. 


CHAPTER  VI. 

THE  PARALLELOGRAM  OF  FORCES. 

38.  IN  the  case  of  the  Inclined  Plane  considered  by  Stevinus 
there  are  three  forces  acting  on  the  sustained  weight:  (1)  its  weight 
pulling  vertically  downwards,  (2)  the  pull  of  the  string  along  the 
slope  of  the  plane,  (3)  the  support  of  the  plane  itself.  This  last 
force  can  be  nothing  but  a  thrust,  or  pressure,  at  right  angles  to 
the  plane ;  for  the  plane  is  supposed  to  be  smooth,  i.e.  incapable 
of  exerting  any  sideway  reaction  on  objects  in  contact  with  it,  of 
the  nature  of  resistance  to  slip. 

Stevinus  not  only  discovered  the  relation  between  the  tension 
and  the  weight,  when  it  was  sustained  on  the  inclined  plane ;  but 
he  perceived  that  the  relation  between  these  three  forces  must  be 
the  same,  if  they  are  to  balance  each  other,  however  they  are 
produced,  provided  that  the  form  of  the  machine's  motion,  i.e.  the 
directions  of  the  forces,  remains  the  same. 


B  C 

Fig.  31.     (From  Mach's  Mechanik.) 


CHAP.  Vl] 


THE   PARALLELOGRAM   OF   FORCES 


49 


For  instance,  the  thrust  R  might  just  as  well  be  replaced  by 
the  tension  of  a  string  perpendicular  to  the  plane,  carried  over  a 
fixed  pulley  at  D,  and  supporting  a  weight  R. 

The  plane  may  now  be  removed,  and  there  is  left  a  so-called 
"  Funicular  Machine." 


Fig.  32. 

Draw   the   vertical   through   a,  and   take   any  length  db   to 
represent  the  weight  W.     From  b  draw  be  perpendicular  to  AB. 
Then  by  Stevinus'  Principle, 


__        _ 
W~  AB~  ab' 
by  similar  triangles. 

But  there  is  no  reason  why  the  function  of  the  two  strings 
should  not  be  interchanged,  and  R  be  supposed  to  support  W  on 
a  plane  DaE,  whose  reaction  is  the  force  P. 
In  this  case 

_R_ad 
W~  ab' 

Hence  the  two  forces  P  and  R,  which  together  balance  W,  are 
represented  by  ac  and  ad,  when  W  is  represented  by  ab  in 
magnitude,  but  of  course  in  the  reversed  direction.  The  line 
ab  must  therefore  represent  the  single  force  which  is  equivalent 
to,  i.e.  has  the  same  effect  as,  P  and  R  acting  together,  since  they 
just  balance  W  acting  downwards. 

c.  4 


50 


MECHANICS 


[CHAP. 


Stevinus  was  thus  led  to  a  special  case  (when  the  two  forces 
are  at  right  angles)  of  a  very  important  proposition,  the  Parallelo- 
gram of  Forces,  which  may  be  stated  as  follows : 

If  two  forces  acting  at  a  point  are  represented  in  magnitude 
and  direction  by  two  straight  lines,  they  are  together  equivalent 
to  a  single  force,  represented  by  that  diagonal  of  the  parallelogram 
constructed  on  the  two  straight  lines  which  passes  through  the 
point. 

This  proposition,  though  employed  by  Stevinus,  was  first 
explicitly  stated  by  Newton  as  a  corollary  to  the  Second  Law  of 
Motion.  It  is  the  starting  point  of  the  modern  treatment  of 
Statics. 

39.  Experiment.  A  direct  experimental  proof  of  the  Paral- 
lelogram of  Forces  is  easily  arranged. 


Fig.  33. 


VI]  THE  PARALLELOGRAM  OF  FORCES  51 

Three  strings  are  knotted  together,  and  provided  with  hooks 
at  the  other  ends.  Two  of  them  pass  over  light  pulleys  and  have 
weights  attached ;  the  third  hangs  vertically,  supporting  a  weight. 

The  directions  of  the  strings,  when  equilibrium  is  attained,  can 
be  traced  on  a  drawing  board  held  behind  them.  The  trace  of  the 
vertical  string  is  produced  upwards,  and  from  any  point  in  it 
parallels  to  the  other  strings  are  drawn. 

It  will  be  found  that  the  sides  and  vertical  diagonal  of  the 
parallelogram  so  constructed  are  always  proportional  to  the  weights 
hanging  from  the  strings  respectively  parallel  to  them. 

We  shall  defer  the  development  of  the  consequences  of  this 
proposition  till  after  it  has  been  deduced  from  the  Laws  of 
Motion. 


4—2 


CHAPTER  VIL 

THE  PEINCIPLE  OF  VIRTUAL  WORK. 

40.  MECHANICAL  problems,  and  especially  the  simple  machines, 
may  be  regarded  from  another  point  of  view.  It  was  first  noted 
by  Stevinus  in  the  case  of  the  pulleys. 

When  a  weight  is  raised  by  means  of  a  cord  passing  over  a 
single  fixed  pulley,  the  "Power"  must  be  equal  to  the  weight,  and 
it  descends  exactly  as  much  as  the  weight  rises. 

By  employing  a  single  moveable  pulley  the  weight  can  be 
raised  with  half  the  Power  (§15);  but  the  cord  to  which  the  power 
is  attached  must  be  pulled  through  twice  the  height  the  weight 
rises. 

If  pulley-blocks  are  used,  with  n  strings  to  the  lower  block, 
the  force  employed  need  only  be  one-nth  part  of  the  weight  (§  16), 
but  since  each  of  the  n  strings  must  be  shortened  as  much  as  the 
weight  rises,  the  end  of  the  power  string  must  move  n  times  as 
far  as  the  weight  is  lifted. 

Stevinus  saw  that  this  principle  applied  to  all  machines,  and 
embodied  it  in  his  phrase : 

"  Ut  spatium  agentis  ad  spatium  patientis,  sic  potentia  patientis 
ad  potentiam  agentis." 

In  other  words,  "  What  is  gained  in  power  is  lost  in  speed." 

'So  that  the  product  of  the  force  exerted  and  the  distance 
moved  through  is  the  same  for  the  Power  as  for  the  Weight. 

Stevinus  shewed  how  to  employ  this  principle  so  as  to  find 
the  relation  between  the  power  and  the  weight  for  complicated 
machines.  Even  when  the  .construction  is  unknown,  it  may  be 
used.  Imagine  (Fig.  33  a)  the  two  handles  A  and  B  to  be 


CHAP.  VII]  THE    PRINCIPLE    OF   VIRTUAL   WORK 


53 


connected  by  any  system  of  mechanism  (levers,  pulleys,  wheel- 
work,  &c.)  enclosed  in  a  box. 

To  find  what  force  will  be 
exerted  by  B  when  a  given  force 
is  applied  to  A,  all  that  is 
necessary  is  to  observe  how  far 
B  moves  for  a  given  movement 
of  A.  Then  the  ratio  of  the 
force  exerted  at  B  to  the  force 
applied  at  A  is  the  same  as  the 
ratio  of  the  distance  moved  by 
A  to  the  distance  moved  by  B. 


Fig.  33  a. 

41.  This  was  the  way  in  which  Galileo  regarded  the  Inclined 
Plane.  He  was  much  occupied  with  the  descent  of  falling  bodies, 
and  was  very  sure  that  heavy  bodies  never  rose  of  their  own 
accord,  but  settled  down  under  gravity  to  the  lowest  place  they 
could  reach. 

If,  then,  no  motion  took  place  on  the  Inclined  Plane,  it  must 
be  for  the  reason  that,  were  the  weights  to  get  into  motion,  there 
could  be  no  rise  or  fall  of  weights  on  the  whole. 

But  when  two  or  more  weights  are  concerned,  some  rising  and 
some  falling,  how  are  we  to  take  into  account  a  large  weight  rising 
through  a  small  distance,  and  perhaps  obliquely,  as  compared  with 
a  small  weight  falling  through  a  large  distance  ?  Galileo  saw  that 
the  essential  factors  were  the  weight  concerned  and  the  vertical 
distance  through  which  it  rose  or  fell ;  and  that  we  should  carry 
to  our  account  the  product  of  the  two. 

Thus  (Fig.  31)  if  W  is  raised  from  the  bottom  to  the  top  of 
the  plane  AB,  it  rises  a  vertical  height  AC,  while  the  Power- 
weight  descends  a  length  equal  to  AB.     Hence,  that  there  may 
be  no  rise  or  fall  of  weights  on  the  whole, 
Px  AB=WxAC, 

P     AC 

W~AB' 


or 


42.     Of  course  Galileo  could  only  have  seen  that  the  vertical 
heights  were  the  essential  factors  from  instinctive  knowledge  based 


54  MECHANICS  [CHAP 

on  experience  ;  and  the  principle  could  only  be  finally  established 
by  careful  comparison  with  experience  according  to  the  canons  of 
logic  (v.  §118). 

"  Galileo's  conception  of  the  Inclined  Plane  strikes  us  as  much 
less  ingenious  than  that  of  Stevinus,  but  we  recognize  it  as  more 
natural  and  more  profound  *." 

When  we  realize  it,  we  suddenly  perceive  how  the  ratio  of  the 
forces  in  the  case  of  the  inclined  plane  fits  in  with  our  general 
experience  that  heavy  bodies  settle  downwards  as  far  as  they  can. 
"The  equilibrium  equation  of  the  principle  may  be  reduced  in 
every  case  to  the  trivial  statement,  that  when  nothing  can  happen 
nothing  does  happen*." 

"  The  principle,  like  every  general  principle,  brings  with  it,  by 
the  insight  which  it  furnishes,  disillusionment  as  well  as  elucidation. 
It  brings  with  it  disillusionment  to  the  extent  that  we  recognize 
in  it  facts  which  were  long  before  known  and  even  instinctively 
perceived,  our  present  recognition  being  simply  more  distinct  and 
more  definite  ;  and  elucidation,  in  that  it  enables  us  to  see  every- 
where throughout  the  most  complicated  relations  the  same  simple 
facts*." 

43.  Torricelli  (1608  —  1647)  gave  the  principle  a  more  general 
form  by  employing  the  notion  of  the  Centre  of  Gravity. 

Let  there  be  a  number  of  weights,  P1?  P2,  &c.  connected  by 
mechanism  ;  and  let  their  heights  above  a  line  of  reference 
Ox  be  h,  /&2,  &c.  Then  the  height  of  the  centre  of  gravity  is, 
(§21) 

,  PA+PA+... 
P.+P.+... 

Let  the  machine  work  for  a  moment  so  that  the  heights  of 
the  weights  become  hft  hz'}  etc.  The  new  height  of  the  centre 
of  gravity  is: 


The  centre  of  gravity  will  have  fallen  a  distance 

,    ,,  A  (A  -  A/)  +  p2(/*2  -/*;)  +  ... 

P1+p,+... 

*  Mach,  Mechanik. 


VIl]  THE    PRINCIPLE    OF   VIRTUAL    WORK  55 

If  such  a  fall  is  possible,  it  will  certainly  take  place.  If 
therefore  the  machine  is  in  equilibrium,  it  must  be  because  the 
centre  of  gravity  of  the  weights  attached  to  it  cannot  descend, 
and  thus 


i.e.  the  sum  of  the  products  of  the  weights  into  their  vertical 
displacements  when  motion  takes  place  must  be  zero. 

44.  In  a  letter  to  Varignon  written  in  1717  John   Bernoulli 
Generalization          shewed  how  to  extend  the  principle  to  all  cases 
of  the  Principle.        of  equilibrium. 

Let  any  number  of  forces  act  in  any  directions  at  any  points. 
Imagine  the  points  to  receive  any  infinitely  small  displacements 
compatible  with  their  mechanical  connections. 

Multiply  each  force  by  so  much  of  the  displacement  of  its 
point  of  application  as  takes  places  along  the  direction  of  the 
force,  counting  the  product  positive  if  the  displacement  occurs  in 
the  same  sense  as  the  force,  and  negative  if  in  the  opposite 
sense. 

Then  in  order  that  there  may  be  equilibrium  the  sum  of  all 
these  products  must  be  equal  to  zero. 

45.  We  can  simplify  this  statement  by  introducing  the  very 

important  term  Work. 

Work. 

In  common  language  any  fatiguing  exertion 
is  called  work.  Lifting  weights  is  a  simple  and  familiar  example. 
Consider  a  number  of  labourers  engaged  in  carrying  bricks, 
mortar,  &c.,  up  vertical  ladders  to  the  different  floors  of  a 
building  in  course  of  construction.  The  amount  of  work  done 
by  any  one  man  depends  on  two  things  :  (1)  the  weight  of  bricks 
lifted  ;  (2)  the  vertical  height  to  which  they  are  raised  ;  for  it  is 
clear  that  the  man  who  lifts  twice  the  weight  of  bricks  to  the 
same  storey  as  another  man,  or  the  same  weight  to  twice  the 
height,  will  have  done  twice  as  much  work. 

And  it  depends  on  these  two  things  only.  It  does  not  depend 
on  the  time  taken  to  do  it.  One  man  may  work  steadily,  but 
slowly  :  another  may  take  frequent  intervals  for  rest  and  refresh- 
ment, and  then  work  furiously.  The  foreman  need  not  watch 


56 


MECHANICS 


[CHAP. 


them ;  he  can  measure  the  work  by  the  piece,  i.e.  by  noting  the 
weight  raised  and  the  height  to  which  it  is  carried. 

Nor  does  it  depend  on  the  path  by  which  the  bricks  are 
carried.  One  man  may  take  them  up  a  vertical  ladder,  by  a  dead 
lift  through  a  short  distance;  another  may  arrange  a  series  of 
sloping  planks,  and  arrive  with  little  effort,  but  after  a  long  walk. 
The  effective  result,  the  work  done,  is  the  same  if  the  same  weight 
is  raised  to  the  same  height. 

Work,  then,  results  from  two  factors, — force  exerted,  and 
distance  through  which  it  moves  its  object.  Both  must  be 
forthcoming  for  work  to  be  done.  Neither  is  sufficient  alone. 
Great  forces  are  often  exerted  without  doing  any  work  in  the 
scientific  sense.  The  piers  of  a  bridge  exert  a  great  upward 
thrust,  but  do  no  work,  though  they  serve  a  useful  end,  for  they 
prevent  gravity  from  doing  work,  and  so  bringing  the  bridge  to  the 
ground. 

46.  If  the  object  moves,  but  not  along  the  direction  of  the 
force,  only  so  much  of  the  displacement  is  to  be  reckoned  as  takes 
place  along  that  direction,  just  as  at  football  it  may  often  be 
advisable  to  run  with  the  ball  obliquely,  but  the  effective  value 
of  the  run  is  estimated  by  the  yards  gained  in  the  direct  line 
between  goals. 


Fig.  34. 

Let  a  curtain  ring  be  pulled  with  force  P  by  a  cord  at  right 
angles  to  the  rod.  No  effect  is  produced ;  no  work  is  done.  Now 
let  the  ring  be  drawn  along  the  rod  by  a  pull  Q,  from  A  to  R 
The  force  Q  does  work,  but  P  does  no  work  in  spite  of  the  motion; 
for  it  has  not  effected  any  advance  of  the  ring  in  its  own  direction. 


VIl]  THE   PRINCIPLE   OF   VIRTUAL   WORK  57 

If  the  same  motion  is  effected  by  applying  the  force  P 
obliquely  (Fig.  34),  P  does  work.  The  effective  distance  through 
which  it  has  moved  the  ring  is  not  AB,  however,  but  AC,  the 
projection  of  AB  upon  the  direction  of  the  force.  This  projection 
has  the  full  value  AB  when  P  acts  along  AB,  and  vanishes  when 
P  is  at  right  angles  to  the  motion. 

47.  In  science  the  term  Work  is  adopted  accordiugly  with  the 
following  definition. 

Work.  A  force  is  said  to  do  work  when  its  point  of  application 
is  displaced  in  the  direction  of  the  force. 

When  the  displacement  is  in  the  opposite  direction,  work  is 
said  to  be  done  against  the  force,  and  is  counted  negative. 

The  unit  of  work  is  the  amount  of  work  done  by  unit  force  in 
displacing  its  point  of  application  through  unit  length. 

If  P  units  of  force  are  acting  through  a  displacement  of 
I  units  of  length,  the  work  done  will  be  P  x  I  units  of  work. 

The  engineer's  unit  of  work  is  the  foot-pound,  i.e.  the  work 
done  in  lifting  one  pound  through  a  vertical  height  of  one  foot. 
If  the  dynamical  unit  of  force  is  employed,  the  unit  of  work  is  the 
foot-poundal  (§  124). 

On  the  C.G.S.  system  the  unit  of  work  is  the  work  done  when 
a  force  of  one  dyne  is  exerted  through  a  distance  of  one  centimetre. 
This  unit  is  called  an  Erg  (§  124). 

48.  Returning    now    to    the    general    principle    stated    by 
Bernoulli,    we   see  that  the  product  of  each  force   by  the  dis- 
placement of  its  point  of  application  along  its  direction  is  the 
work  done  by  the  force  during  the  displacement,  and  is  to  be 
counted  positive  or  negative  according  as  the  point  moves  in  the 
direction  in  which  it  is  urged  by  the  force,  or  the  opposite.     And 
since  the  work  is  not  really  done  (for  the  system   remains   in 
equilibrium,  and  the  imaginary  displacements  are  only  an  artifice 
to  enable  us  to  perceive  the  relations  between  the  forces  required 
to  maintain  equilibrium)  the  word  virtual  is  used  to  indicate  that 
both  displacements  and  the  consequent  work  done  are  such  as 
might  occur,  consistently  with  the  structure  of  the  system. 

With  these  conventions  the  principle  of  Virtual  Work  may  be 
stated  succinctly  as  follows : 


58 


MECHANICS 


[CHAP. 


A  system  of  forces  will  be  in  equilibrium,  if  the  total  virtual 
work  for  any  infinitely  small  displacements  consistent  with  the 
conditions  is  zero. 

49.  The  displacements  are  to  be  taken  infinitely  small,  for  if 
a  finite  motion  is  allowed,  the  system  may  pass  over  into  some 
other  configuration,  where  different  conditions  of  equilibrium  may 
prevail. 


Fig.  35. 

This  is  not  always  the  case.  Thus  in  Fig.  35,  the  relation 
between  the  weights  P  and  W  will  be  the  same  so  long  as  W 
remains  at  D,  whether  it  be  supported  by  the  sphere  or  the  plane 
which  touches  the  sphere  at  D. 

(1)  Let  it  be  on  the  plane.  Suppose  the  weight  P  to  descend 
till  W  arrives  at  D'.  Then  the  vertical  rise  of  W,  G'D\  is  to  the 
vertical  fall  of  P,  DD',  as  BG  to  AB,  i.e., 

C'D'  =  DD'smA, 

whether  DD'  be  small  or  great.     The  equation  of  virtual  work, 
W  x  DD'  sin  A  —  P  x 

ie.  —  =  BC 


VII] 


THE    PRINCIPLE   OF   VIRTUAL   WORK 


59 


holds  good  equally  well  for  an  infinitesimal  displacement,  or  for  the 
whole  length  of  the  plane. 

(2)  But  if  the  weight  is  resting  on  the  sphere,  it  is  only  for  an 
infinitesimal  displacement  (i.e.  for  so  long  as  the  sphere  may  be 
taken  to  coincide  with  its  tangent)  that  we  get  P/TF=sin-4. 
Farther  along  the  sphere  the  inclination  of  the  tangent  and  the 
ratio  P/W  are  different,  so  that  if  we  want  the  conditions  of 
equilibrium  at  D,  we  must  restrict  ourselves  to  an  infinitely  small 
displacement. 

We  shall  now  apply  this  principle  to  a  few  cases  that  are  not 
so  easy  of  solution  by  other  means. 


50. 


The  Screw. 


This  is  only  a  case  of  the  Inclined  Plane,  as  may  be  seen 
by  cutting  out  a  right-angled  triangle  in  paper 
and  wrapping  it  round  a  ruler. 
The  Power  is,  however,  applied  parallel  to  the  base  of  the  plane, 
as  in  the  Wedge ;  and  the  plane  is  made  to  slide  under  the  weight 
so  as  to  raise  it. 

The  distance  from  thread  to  thread  of-  the  screw,  measured 
parallel  to  the  axis,  i.e.  the   distance  through  which  the  screw 


Fig.  36. 

advances  for  one  turn  of  the  head  or  lever  arm,  is  called  the  Pitch 
of  the  screw. 


60 


MECHANICS 


[CHAP. 


Let  the  pitch  of  the  screw  be  p,  and  the  length  of  the  lever- 
arm  I. 

As  in  the  inclined  plane,  the  form  of  the  mot:on,  and  hence 
the  law  of  equilibrium,  remains  unchanged  however  far  the  screw 
be  turned.  We  need  not  therefore  restrict  ourselves  to  an  infinitely 
small  motion.  Let  us  suppose  the  screw  to  make  one  complete 
turn. 

Then,  apart  from  friction,  the  virtual  work  consists  of  P  x  2?rZ 
done  by  the  Power ;  and  W  x  p  done  against  the  Weight.  Thus 
for  equilibrium, 


and 


P_       p    _  pitch 

W     2irl     circumference  of  power-circle 


51.     Weston's  Differential  Pulley  consists  of  an  upper  block 

TheDifferen-  With      tWO      grOOVCS      of 

slightly  different  radii, 
R,  r,  connected  by  an  endless  chain, 
as  in  the  figure,  with  a  single  move- 
able  pulley.  The  grooves  of  the  upper 
block  contain  notches  or  teeth  which 
fit  into  the  links  of  the  chain  so  that 
it  cannot  slip.  . 

Here  again  we  may  take  the  dis- 
placement large  if  we  wish.  Let  the 
upper  block  make  one  revolution.  The 
virtual  work  done  by  the  Power  is 

P  X  27T.R. 

Meanwhile  the  weight  is  raised  by 
half  the  difference  between  the  length 
of  chain  wound  up  on  the  large  groove 
of  the  upper  block,  and  that  which  is 
unwound  from  the  small  one.  Hence 
the  virtual  work  done  against  the  weight  is 


Fig.  37. 


Wx 


277-jR  -  277T 


Vll]  THE   PRINCIPLE   OF   VIRTUAL   WORK 

For  equilibrium  : 


61 


P     R-r 
W  =  ''   2R   ' 

52.     A  common  form  of  balance  for  weighing  letters  is  that 
Robervai-s          devised  by  Roberval. 
Balance-  The  scales  are  attached  to  the  two  vertical 


Fig.  38.     Roberval's  Balance. 


Fig.  39. 


62  MECHANICS  [CHAP. 

sides  of  a  jointed  parallelogram,  the  other  two  sides  turning  about 
pins  at  their  centres. 

If  the  balance  moves,  one  scale  descends  exactly  as  much  as 
the  other  rises.  Hence  it  does  not  matter  where  the  weights  are 
placed  on  the  scales.  If  they  are  equal,  their  virtual  works  are 
equal  and  opposite,  and  there  will  be  equilibrium. 

53.     In  the  form  of  corkscrew  figured  on  the  preceding  page 
the  handle  moves  four  times  as  far  as  the  head  of 
the  screw.     Hence  the  pull  exerted  on  the  screw 
is  four  times  as  great  as  that  applied  to  the  handle.  ' 


EXAMPLES. 

(Many  problems  may  be  solved  very  easily  by  the  principle  of  Virtual 
Work  when  the  Differential  Calculus  is 
employed  to  find  the  small  imaginary 
displacements  of  the  parts  of  the  system. 
For  example  : 

A  hinged  parallelogram  of  sides  a,  b 
has  its  opposite  corners  joined  by  strings 
screwed  up  to  tensions  T  and  T'.  Find 
one  angle  of  the  parallelogram. 

Suppose  the  parallelogram  distorted 
so  that  the  angle  o>  is  slightly  increased 

tO  O)  +  d(O. 

The  reactions  at  each  hinge  are  equal  and  opposite  for  the  two  rods  meeting 
there,  so  that  their  virtual  work  vanishes. 
The  sum  of  the  v.w.'s  for  T  and  T'  is 


Now 


AC.d(AC)=  -  ab  sin  a>  .  da, 
££.d(£D)=+absma>.da>. 

T        d(BD}_AC 
'   T'~     d(AC)~BD' 


whence 


As  examples  which  may  be  solved  without  the  Calculus  take  the  following  : 


VII] 


THE    PRINCIPLE   OF   VIRTUAL    WORK 


1.  A  light  wire  is  stretched  over  two  smooth  pulleys  at  a  distance  of 
10  feet  from  each  other  in  the  same  horizontal  line,  and  has  1121bs.  hung  at 
each  end.     What  weight  hung  at  the  middle  of  the  wire  will  cause  it  to  sag 
one  inch  ? 

2.  An  elastic  ring  of  natural  length  I  and  weight  W  is  laid  over  a  smooth 
vertical  circular  cone  of  angle  2a.     The  tension  of  the  ring  when  stretched  to 

I'  —  I 

a  length  I'  is  given  by  T=  T0  —j— .     At  what  depth  below  the  vertex  will  the 

I 

ring  rest  ? 

54.     Let  any  forces  P,  Q,  R,  &c.,  act  at  points  A,  B,  C,  &c., 

and  suppose  them  to  be  applied  as  follows.     Let 

praoofaonfgthe  there   be  pulleys  at  A,  B,  C,  with   other   fixed 

principle  of  virtual     puiieys   in   the   proper   directions  at  A',  B',  C'. 

Attach  a  string  at  A',  and  carry  it  P  times  back 

and  forth  to  A ;  then  round  B',  and  Q  times  back  and  forth  to  B, 


Fig.  40. 

and  so  on  for  all  the  forces.     Finally  let  it  hang  from  the  last 
pulley,  and  attach  a  weight  equal  to  half  a  unit.     Then  since  the 


64  MECHANICS  [CHAP. 

tension  is  the  same  throughout,  if  the  pulleys  be  smooth,  and 
2P,  2Q,  2R,  &c.  strings  run  to  A,  B,  C,  respectively,  the  forces 
P,  Q,  R,  will  be  applied  at  those  points. 

Now  if  among  the  possible  mutual  displacements  of  the  points 
A,  B,  C,  &c.,  there  be  any  which  would  on  the  whole  allow  the 
half-unit  weight  to  descend,  then  it  certainly  will  descend,  and 
work  will  be  done.  But  if  the  weight  remained  at  the  same  level, 
or  had  to  rise,  whatever  combination  of  small  movements  were 
given  to  A,  B,  G,  then  motion  would  not  ensue. 

Suppose  that  the  result  of  any  such  movements  of  A,  B,  C, 
were  to  shorten  the  strings  between  A  A'  by  an  amount  a,  those 
between  BB'  by  6,  and  so  on.  Then  for  equilibrium  the  total 
shortening,  i.e.,  the  amount  by  which  the  last  weight  would 
descend,  must  be  zero,  or  less  than  zero. 

But  the  shortening  of  the  2P  strings  at  A  is  2Pf,  and  so  for 
the  others.  Thus 


or 

The  sum  of  the  virtual  works  is  therefore  zero  or  negative. 

55.  Lagrange's   ingenious   idea   makes   it   easier   for    us    to 
understand  the  principle,  for  it  enables  us  to  fix  our  attention 
on  the  motion  of  one  weight  instead  of  many.     But  it  is  not  a 
proof  that  the  possibility  or  impossibility  of  doing  work  is  decisive 
of  equilibrium.     That  principle  is  involved  in  each  of  the  pulleys 
he  employs,  as  much  as  in  the  more  complicated  system.     It  can 
only  be  derived  from  experience. 

56.  Lagrange's  arrangement  also  helps  us  to  study  the  system 
of  bodies  A,  B,  0,  acted  on  by  forces  P,  Q,  R,  regarded  as  a 
machine  for  doing  work.     Let  the  hanging  weight  carry  a  pencil 
pressing  against  a  sheet  of  paper,  carried  past  it  horizontally,  — 
wound,  for  instance,  on  a  drum  with  vertical  axis.     Then  if  the 
system   be   allowed   to   move   (consistently  with   its   mechanical 
connections)  the  depth  of  the  hanging  weight  below  its  original 
position  will  be  an  indicator  of  the  work  done  by  the  system  in 
reaching  any  other  configuration.      The  pencil  will  record  this 
depth  in  a  curve,  as  in  Figure  41. 


VII] 


THE    PRINCIPLE   OF   VIRTUAL   WORK 


65 


It  was  pointed  out  by  Maupertuis  in  1740  that  when  the 
system  arrives  at  a  position  of  equilibrium,  the  work  done  is 
in  general  a  maximum  or  a  minimum.  The  weight  is  at  a 
turning  point  of  the  curve. 


1 

b 

Fig.  41. 

When  its  height  is  a  maximum,  as  at  a,  c,  the  system  can  do 
work  (the  weight  can  descend),  if  disturbed  from  the  position  of 
equilibrium  on  either  side  of  it.  But  when  the  height  is  a 
minimum,  as  at  6,  the  system  can  only  do  work  by  returning  to 
the  position  of  equilibrium  if  disturbed  from  it. 

Stable  equilibrium  therefore  corresponds  to  a  maximum  of 
work  done  by  the  system,  unstable  equilibrium  to  a  minimum. 

If  the  curve  remains  horizontal  for  any  finite  distance,  as 
at  d,  e,  equilibrium  exists  for  all  the  corresponding  positions, 
with  no  tendency  to  pass  from  one  to  another.  The  equilibrium 
is  then  neutral  equilibrium,  as  when  a  sphere  rests  on  a  horizontal 
plane. 


c. 


CHAPTER  VIII. 

REVIEW  OF  THE  PRINCIPLES  OF  STATICS. 

57.  IN   the   historical   development  of  Statics  the   different 
investigators   have   adopted   different   tests  for  the  existence  of 
equilibrium. 

Archimedes  fixed  his  attention  on  the  weights  and  their 
distances  from  a  fulcrum,  and  arrived  at  the  principle  of  the  Lever. 

Stevinus  divined  the  principle  of  the  Inclined  Plane,  and 
referred  equilibrium  to  a  relation  between  the  forces  and  their 
directions,  more  fully  expressed  in  the  Parallelogram  of  Forces. 

Galileo  saw  that  equilibrium  was  determined  by  the  weights 
and  their  vertical  descent  towards  the  earth,  and  so  reached  the 
principle  of  Work. 

Each  of  these  principles  is  an  expression  of  our  experience 
from  one  point  of  view  or  another.  As  such  they  are  equally 
valid.  Their  authority  is  coequal,  and  each  is  sufficient  in  itself 
as  a  foundation  for  the  science  of  Statics.  We  may  develope 
it  from  any  one,  or  employ  them  all.  Which  of  them  shall  be 
selected  is  a  matter  of  convenience,  or  of  historical  accident. 

58.  As  we  might  expect,  they  are  mutually  deducible. 

The  Parallelogram  of  Forces  has  already  been  deduced  from 
the  Inclined  Plane  (§  38),  at  all  events  for  the  case  when  the 
forces  are  at  right  angles. 

Galileo  deduced  the  Inclined  Plane  from  the  Lever. 

He  points  out  that  the  ratio  of  P  to  W  depends  on  the  form  of 
the  motion,  i.e.  that  W  should  move  along  aB,  while  P  descends 
vertically.  It  is  a  matter  of  indifference  whether  W  is  compelled 


CHAP.  VIII]  REVIEW   OF   THE    PRINCIPLES   OF    STATICS 


67 


to  do  this  because  it  rests  on  a  plane  AB,  or  for  some  other  reason, 
as,  for  instance,  that  it  should  be  attached  by  a  bar  aO  to  a  fixed 


Fig.  42. 

pivot  at  0.     It   would  still  have  to  begin  to  move  along  aB,  at 
right  angles  to  Oa. 

But  in  this  case  by  Leonardo's  form  of  the  principle  of  the 
Lever  (§  11), 

PxOa=Wx  Ob, 

P_0&_JSC 
W~  Oa'AB' 

We  shall  deduce  the  Principle  of  the  Lever  from  the  Parallelo- 
gram of  Forces  later  (§  174). 

The  Principle  of  the  Inclined  Plane  has  been  deduced  from 
that  of  Work  (§  40). 

The  mutual  relation  of  the  Principle  of  the  Lever  and  that  of 
Work  is  easily  seen. 


Fig.  43. 


5—2 


68  MECHANICS  [CHAP.  VIII 

Let  the  Lever  ACB  receive  an  infinitely  small  displacement  to 
a  new  position  A'CB'. 

The  virtual  work  done  by  P  =  P  x  A  A'. 
The  virtual  work  done  against  W  =  W  x  BB'. 
Assuming  the  Principle  of  Virtual  Work,  we  have 
P  x  AA'  -  W  x  BB'  =  0. 

P  _  BB'  _  BC 
W~  AA'~  AC' 

But  this  is  the  Principle  of  the  Lever.  The  converse  is 
obviously  true. 

59.  It  is  interesting  to  trace  in  this  manner  the  connection 
between  the  various  principles,  but  it  does  not  increase  the 
authority  of  any  one  of  them  to  deduce  it  from  another.  Having 
followed  the  actual  historical  order  in  which  they  were  arrived  at, 
we  are  at  liberty  to  make  any  one,  or  all  of  them,  the  starting 
point  of  further  developments.  The  modern,  and  on  the  whole 
the  most  convenient,  practice  is  to  deduce  the  Parallelogram  of 
Forces  from  Newton's  Laws  of  Motion,  themselves  but  another 
expression  of  experience.  The  other  principles,  and  the  whole 
science  of  Statics,  can  then  be  built  on  this  principle.  Statics 
thus  becomes  a  special  case  of  Dynamics,  when  the  forces  concerned 
happen  to  be  in  equilibrium.  This  is  the  course  we  shall  now 
adopt,  leaving  the  farther  development  of  Statics  till  we  have 
traced  the  discovery  of  the  fundamental  principles  of  Dynamics. 


CHAPTER  IX. 

GALILEO  AND  THE  BEGINNINGS  OF  DYNAMICS. 

60.  IN  1638,  when  Stevin  had  already  cleared  up  so  much  of 
The  Problem  of  Statics,  no  progress  had  been  made  with  that  part 
Failing  Bodies.  o£  ^e  subject,  now  called  Dynamics,  which  deals 
with  Motion.  The  first  problem  to  be  considered  was,  naturally, 
the  familiar  case  of  the  fall  of  heavy  bodies  to  the  earth.  Its 
solution  was  the  achievement  of  Galileo,  who  in  the  course  of  his 
researches  was  led  to  the  discovery  of  several  principles  of  general 
importance  in  Mechanics. 

It  is  not  easy  at  the  present  day  to  realize  the  difficulties 
Galileo  had  to  encounter.  Let  us  try  to  strip  ourselves  of  what  is 
now  common  knowledge,  and  see  what  were  the  views  held  in  his 
day,  with  all  the  authority  of  two  thousand  years'  acceptance 
backed  by  the  great  name  of  Aristotle. 

The  fall  of  heavy  bodies  (and  the  rise  of  light  bodies  which 
often  accompanied  it)  was  accounted  for  by  assuming  that  "  every 
body  sought  its  natural  place/'  and  that  the  place  of  heavy  bodies 
was  below,  that  of  light  bodies  above. 

Thus  in  the  Elzevir  edition  of  Stevin,  Leyden  1634,  the  editor, 
Albert  Girard,  speaks  of  "  Tant  de  millions  de  matieres,  qui  sont 
disposees  chacunes  en  leurs  lieux,"  and  gives  a  general  definition 
of  gravity. 

"  Pesanteur  est  la  force  qu'une  matiere  demonstre  &  son 
obstacle,  pour  retourner  en  son  lieu." 

"  Ce  que  je  demonstreray,  et  soustiendray  en  temps  et  lieu,  a 
ceux  qui  ne  le  pourront  pas  comprendre." 

When  movements  were  observed  in  which  heavy  bodies  rose 
and  light  ones  fell  for  a  time,  such  motions  were  distinguished 


70  MECHANICS  [CHAP. 

from  "natural"  motions  by  the  term  "violent."     It  was  believed 
that  heavy  bodies  fell  more  quickly  than  light  ones. 

It  will  be  seen  that  such  ideas  were  too  vague  to  serve  as 
starting  points  for  progress.  They  were  guesses  at  the  reason  why 
bodies  fell ;  attempts  to  find  a  cause  for  their  motion. 

61.  It  was  already,  as  Mach  has  pointed  out,  a  proof  of  genius 
that  Galileo  could  so  far  shake  himself  free  from  the  prevailing 
notions  of  his  time,  as  to  take  up  the  modern  point  of  view, 
and  ask  himself  first  how  bodies  fell.     That  is  to  say,  he  began 
by  investigating  the  facts,  and  tried  to  discover  the  rule  or  law 
according  to  which  the  fall  took  place. 

Now  a  falling  body  starts  from  rest  and  passes  over  a  certain 
distance  in  a  certain  time  with  a  speed  which  a  very  slight 
observation  shews  to  be  rapidly  increasing.  The  questions  which 
Galileo  thought  important  were  such  as  these :  How  is  the  speed 
acquired  by  the  body  in  its  fall  related  to  the  distance  it  has 
fallen  ?  Or  to  the  time  of  fall  ? 

Here  again  he  was  met  by  difficulties,  in  the  lack  of  experi- 
mental means  for  measuring  times  and  speeds.  The  mechanical 
clocks  of  his  day  were  useless  except  for  considerable  lengths  of 
time,  and  could  not  be  relied  on  for  measuring  a  few  seconds 
or  fractions  of  a  second.  One  is  at  a  loss  to  know  whether  to 
admire  more  the  ingenuity  with  which  he  overcame  the  experi- 
mental difficulties,  or  his  philosophical  insight  as  to  the  real  points 
to  be  investigated. 

In  his  treatise,  Discorsi  e  Dimonstrationi  Matematici,  he 
begins  by  a  guess  which  seems  natural  enough  at  first  sight,  that 
the  speed  acquired  will  be  proportional  to  the  distance  the  body 
has  fallen  from  rest.  But  before  putting  this  to  the  test  of 
experiment,  he  examines  the  hypothesis,  and  convinces  himself 
that  such  a  rule  of  motion  involves  a  contradiction,  is  in  fact 
inconsistent  with  itself. 

62.  The  next  idea  that  occurs  to  him  is  that  perhaps  the 
speed  will  be  proportional  to  the  time  of  fall.     Finding  no  con- 
tradiction in  this,  he  proceeds  to  test  it  experimentally. 

Since  it  was  next  to  impossible,  with  the  means  at  his  disposal, 


IX] 


GALILEO   AND   THE    BEGINNINGS   OF   DYNAMICS 


71 


to  measure  the  speed  acquired  by  a  body  even  in  a  short  fall,  he 
calculates  the  distance  that  a  body  ought  to  fall  through,  on  the 
hypothesis  that  the  speed  acquired  was  at  each  instant  proportional 
to  the  time  elapsed  from  the  start.  His  proof  is  a  good  instance 
of  the  use  of  graphic  methods. 


o 


Fig.  44. 


63.  Let  us  represent  the  time  elapsed  by  a  straight  line  OA, 
which  may  be  divided  up  so  as  to  represent  the  different,  intervals 
of  which  the  whole  time  is  composed.  At  each  point,  such  as  D, 
erect  a  perpendicular  whose  length  shall  be  proportional  to  the 
speed  acquired  at  the  moment  D.  Then,  since  the  speed  increases 
proportionally  to  the  time,  the  ends  of  these  perpendiculars  will 
lie  along  the  straight  line  OB. 

Let  OP  be  the  speed  at  the  middle  moment.  It  is  clearly 
half  AB,  the  final  speed. 

Consider  two  points,  D,  E  equidistant  from  C.  The  speeds 
DQ,  ER  will  be  the  one  as  much  less,  as  the  other  is  greater  than 
CP,  the  speed  at  the  middle  moment.  So  that  if  we  compare  the 
real  motion  with  that  of  a  body  which  should  start  with  the  speed 
CP  and  maintain  it  unchanged  throughout,  we  see  that  any  loss 
of  distance  travelled,  owing  to  the  smaller  speed  at  D,  will  be 
exactly  compensated  for  by  the  greater  speed  at  the  corresponding 
point  E.  The  distances  fallen  through  by  the  two  bodies  will 
therefore  be  the  same  in  the  end. 


72  MECHANICS  [CHAP. 

As  we  do  not  know  the  speed  acquired  by  a  falling  body  during 
a  fall  of  one  second,  let  us  call  it  g  feet  per  second.  Then  if  the 
idea  that  the  speed  is  proportional  to  the  time  of  fall  be  correct, 
the  speed  at  the  end  of  t  seconds  will  be  gt.  The  speed  at  the 
middle  moment  will  be  gt/2,  and  the  distance  fallen  by  a  body 
moving  with  this  speed  unchanged  for  the  whole  t  seconds,  will  be 

«?-x;*-£ 

2  *      ~  2  ' 

We  could  thus  make  a  table  for  different  numbers  of  seconds, 
as  follows : 


Time  of  Fall 

Speed  acquired 

Space  fallen 

1  second 

9 

$r/2xl  =  g/2 

2 

2g 

g/2  x  22  =  2g 

3 

% 

g/2x&  =  9g/2 

*  gt  g/2xt* 

64.     Galileo's  Method. 

To  avoid  the  difficulties  introduced  by  the  great  speed  acquired 

Experimental         ^J   a   body   falling   freely  even   for  one   or   two 

seconds,  Galileo  assumed  that  a  ball  rolling  down 

an  inclined  plane  in  a  groove  would  follow  the  same  kind  of  rule 

as  a  freely  falling  body,  but  with  diminished  speed. 

Marking  the  groove  at  different  distances  from  the  top,  he 
proceeded  to  measure  the  times  occupied  by  the  ball  in  reaching 
the  various  marks,  and  verified  that  the  distances  travelled  really 
increased  as  the  squares  of  the  times. 

For  the  measurement  of  the  times  Galileo  made  an  ingenious 
modification  of  the  water-clock  of  Archimedes,  which  had  not  been 
hitherto  applied  to  the  measurement  of  small  times.  The  speed 
at  which  water  flows  out  of  a  hole  in  the  bottom  of  a  vessel 
depends  on  the  height  of  water  standing  above  the  hole.  Galileo 
took  a  broad  vessel  of  large  area,  and  hence  the  level  was  not 
appreciably  altered  during  one  of  his  experiments.  At  the 
moment  when  the  ball  was  released  he  removed  his  finger  from 
the  hole,  allowing  the  water  to  flow  into  a  vessel  which  was  placed 
on  a  balance.  When  the  ball  reached  a  mark,  the  hole  was  closed 


IX] 


GALILEO   AND  THE   BEGINNINGS  OF  DYNAMICS 


73 


by  the  finger  again,  and  the  time  elapsed  could  be  measured  by 
weighing  the  water  which  had  escaped. 


Figure  45  represents  a  modern  version  of  Galileo's  apparatus. 
It  is  found  that  the  squares  of  the  times  required  to  reach  the 
different  marks  are  proportional  to  the  distances  of  the  marks 


MECHANICS 


[CHAP. 


from  the  starting  point.  The  student  should  make  experiments 
with  different  slopes  of  the  plane,  and  plot  the  results  on  squared 
paper,  laying  out  the  distances  along  a  horizontal  line,  and  the 
times  in  the  vertical  direction.  The  curve  so  obtained  will  be  a 
parabola.  If  we  choose  the  vertical  ordinates  to  represent  the 
squares  of  the  times,  instead  of  the  times  themselves,  we  obtain  a 
straight  line. 


\ 

^ 

cc 

^ 

y 

i 

/* 

r 

g 

i  ; 

/ 

z 

ce 

/ 

s 

A 

I 

/ 

/ 

/ 

Di 

St^ftU. 

e  i. 

n, 

7m. 

t. 

100 


200 


1 

/ 

7 

• 

2 

5 

/! 

.c 

/ 

1 

5 

/ 

1 

B 

/ 

«H 

I/ 

u 

i 

1 

i. 

/ 

1 

/ 

™ 

; 

/ 

7 

/ 

t 

/ 

2 

DI,. 

tC&7lC&     I 

n- 

J7771*. 

3                       100                    200                    300 

Fig.  46. 


Fig.  47. 


65.  With  modern  apparatus  it  is  easy  to  verify  the  same  law 
for  bodies  falling  freely. 

Experiment.  Let  a  plate  of  smoked  glass  be  suspended  as  in 
the  figure,  so  that  when  the  thread  is  burnt  away,  it  will  fall  past 
a  horizontal  tuning  fork  to  which  a  bristle  is  attached.  The  fork, 
when  sounded,  makes  a  definite  number  of  vibrations  every  second, 
say  100.  The  result  will  be  a  trace  on  the  smoked  glass  con- 
sisting of  a  number  of  waves  growing  longer  and  longer,  since,  as 
the  plate  gathers  speed,  a  greater  and  greater  distance  will  be 
travelled  in  each  hundredth  of  a  second. 

By  measuring  the  length  of  a  wave  at  any  distance  from  the 
starting  point,  we  can  find  the  speed  at  which  the  plate  was 
moving  after  falling  through  that  distance.  Make  several  such 


IX] 


GALILEO    AND   THE    BEGINNINGS   OF    DYNAMICS 


75 


measurements.     Since  the  distance  fallen  is  proportional  to  the 
square  of  the  time,  and  the  speed  proportional  to  the  time,  the 


Fig.  48. 


squares  of  the  speed  will  vary  as  the  distances  fallen  through. 
Plot  your  results  on  squared  paper,  and  see  if  this  is  the  case. 


between  Speed 
acquired  and 
Vertical  Fall. 


Ideas,  of  great  importance  in  Mechanics,  suggested  to  Galileo 
by  his  experiments. 

66.  In  default  of  means  for  studying  directly  the  motion  of  a 
.  The  connection  body  falling  freely,  Galileo  asked  himself  how  the 
motion  of  a  body  sliding  down  an  inclined  plane, 
would  be  related  to  that  of  a  body  falling  freely. 
He  concludes  that : 

The  speed  attained  on  an  inclined  plane  must  be  the  same  as 
that  attained  in  falling  freely  through  the  same  vertical  height. 

At  first  sight  this  seems  a  startling  assumption.  But  consider 
what  would  be  the  consequence  if  it  were  not  true. 

Let  a  body  slide  from  A  to  B,  and  then  be  reflected  up  the 
equally  inclined  plane  EG  (Fig.  49).  Galileo  feels  that  it  will 
exactly  reverse  its  motion,  losing  speed  precisely  as  it  gained  it 


76 


MECHANICS 


[CHAP. 


along  AB,  and  coming  to  rest  at  C,  at  the  height  from  which  it 
started. 


Fig.  49. 

Let  it  next  be  reflected  up  a  plane  of  less  inclination,  BC'.  It 
must  still  reach  the  same  height.  For  if  it  went  farther,  it  would 
have  risen,  without  external  aid,  beyond  its  original  height ;  and 
by  arranging  a  series  of  steep  and  gentle  slopes  alternately  we 
could  make  a  body,  starting  from  the  top  of  the  first  slope,  raise 
itself  unaided  to  any  height  we  choose.  But  this  we  feel  to  be 
contrary  to  all  our  experience.  "We  have  never  met  with  anything 
like  it.  If  on  the  other  hand  it  failed  to  reach  C',  we  should  only 
have. to  start  the  body  from  C'  and  it  would  rise  above  A,  and  the 
same  contradiction  of  experience  would  arise. 

It  must  therefore  rise  to  C',  neither  more  nor  less.  Hence  the 
speed  attained  must  be  the  same  for  the  same  vertical  fall,  what- 
ever the  slope  of  the  plane  on  which  it  takes  place. 


IX]  GALILEO    AND   THE   BEGINNINGS   OF   DYNAMICS  77 

67.  It  is  characteristic  of  Galileo's  modern  spirit  that  he  at 
once  proceeds  to  test  this  conclusion  by  an  experiment,  and  that  a 
very  beautiful  one. 

Experiment.  Hang  a  bullet  by  a  thread  from  a  nail  0,  and 
drawing  it  aside  to  A  release  it.  It  describes  the  curve  AB,  which 
Galileo  perceives  may  be  regarded  as  a  series  of  very  short  inclined 
planes  of  different  slopes,  so  that  his  law  should  hold  for  the  curve 
as  well  as  for  the  plane. 

It  then  reverses  its  motion,  arriving  at  C,  in  the  same  hori- 
zontal level  with  A. 

Drive  a  nail  in  at  E,  and  repeat  the  experiment.  After 
passing  the  vertical  the  bullet  will  describe  a  circle  of  shorter 
radius  BD.  But  it  will  rise  to  exactly  the  same  level,  whatever 
circle  we  make  it  describe. 


68.  Galileo's  assumption  no  longer  strikes  us  as  strange  when 
we  realize  that  it  involves  only  the  perception  that  bodies  cannot, 
unaided,  raise  themselves  to  a  higher  level  above  the  earth,  a  fact 
which  we  feel  instinctively  to  agree  with  all  our  experience. 

With  the  aid  of  this  assumption,  since  he  could  already  deter- 
mine the  space  fallen  through  in  a  given  time  on  an  inclined  plane, 
he  was  able  to  form  a  notion  of  the  velocity  acquired  by  a  body 
falling  freely.  Let  us  however  perform  the  experiment  directly. 

69.  Experiment     Apparatus  required : — A  pendulum  ticking- 
seconds  loudly  ;  or  a  metronome  set  to  beat  seconds  or  half  seconds. 
A  ball  which  may  be  dropped  from  a  height  of  16*1  feet. 

If  the  ball  be  released  by  hand  precisely  at  any  tick  of  the 
pendulum,  it  will  be  heard  to  strike  the  floor  at  the  next  tick. 
(The  result  will  be  more  accurate  if  the  ball  be  of  iron,  suspended 
from  an  electromagnet  whose  circuit  is  broken  by  the  pendulum 
itself  as  it  makes  the  first  tick.  Experimental  Mechanics.  Sir  R. 
Ball.) 

Since  the  ball  covers  161  feet  in  one  second,  its  average  speed 
must  be  16'1  feet  per  second.  But  this  (§  63)  is  one-half  the  final 
speed.  Therefore  in  one  second  it  has  acquired  a  speed  of  32'2  feet 
per  second. 


78  MECHANICS  [CHAP. 

70.  We  must  now  attend  to  an  inference  which  Galileo  draws 
2.  The  First  Law  fr°m  his  experiments,  as  it  were  incidentally,  and 
of  Motion.  probably  without  seeing  its  full  importance.  He 

was  most  likely  led  to  it  by  the  Principle  of  Continuity.  In 
geometry  a  doubtful  conclusion  may  often  be  tested  by  trying 
whether  it  is  true  in  an  extreme  case.  The  great  investigators 
have  this  principle  continually  in  mind.  "  Natura  non  agit  per 
saltum."  It  is  not  likely  that  one  law  should  hold  good  in  one 
case,  and  quite  a  different  one  in  some  other  case  only  slightly 
different  from  it.  Apply  this  to  the  case  of  the  body  falling  down 
the  inclined  plane  in  figure  49.  What  will  happen  if  the  second 
plane  be  made  more  and  more  oblique  ?  It  is  clear  the  body 
will  have  to  travel  farther  and  farther  before  it  reaches  the  level 
AC',  and  if  the  second  plane  be  made  ultimately  horizontal,  the 
body  will  never  reach  the  level,  however  far  it  may  travel.  But 
in  this  case  it  would  go  on  for  ever !  At  the  same  time  we  see  that 
no  part  of  the  weight  is  employed  in  stopping  it.  Hence  a  motion 
once  started,  will  continue  indefinitely  if  nothing  interferes  to  stop  it. 

Now  this  is  an  entirely  new  point  of  view,  not  only  contrary 
to  the  current  ideas  of  Galileo's  time,  but  surprising  to  many 
uninstr acted  people  even  at  the  present  day.  Since  all  the  move- 
ments we  observe  in  practice  are  shortly  brought  to  rest  by 
various  frictions  and  resistances,  it  is  quite  natural  to  imagine,  as 
the  unobservant  do  to  this  day,  that  every  motion  requires  some 
cause  or  force  to  maintain  it,  and  ceases  when  the  force  is  with- 
drawn. Logic  will  not  help  us  to  settle  the  question.  For 
against  the  principle,  "  the  effect  of  a  cause  persists/'  we  may  set 
another,  such  as  "cessante  causa  cessat  effectus."  Nothing  but 
experiment  can  decide  whether  in  this  case  the  "effect"  of  the 
force  is  the  change  of  place,  or  the  speed  acquired.  Now  Galileo 
perceives  from  his  experiments  that  the  "  effect "  of  the  weight  of 
a  body  is  to  produce  a  change  in  its  speed*. 

We  shall  see  what  an  advance  this  was,  if  we  compare  the  old 
idea  that  "  bodies  sought  their  place,"  that  of  heavy  bodies  being 
below,  with  Galileo's  notion.  When  a  stone  is  thrown  upwards, 
the  first  principle  seems  to  be  contradicted,  for  the  heavy  stone 
rises.  But  Galileo  sees  that  its  weight  is  changing  its  speed,  as 

*  See  Mach,  Mechanics,  pp.  140—143. 


IX]  GALILEO   AND   THE   BEGINNINGS   OF   DYNAMICS  *79 

much  during  the  rise  as  during  the  fall,  reducing  it  by  32'2  feet 
per  second  in  every  second,  till  at  last  it  comes  to  rest,  and  then 
begins  the  descent. 

71.     The  notion  of  speed  as  applied  to  bodies  moving  uni- 

3.   variable         formly  must   have   been  as   familiar  in  Galileo's 

velocity.  day  as  in  ours.     If  a  man  walks  steadily  for  three 

hours  and  finds  he  has  travelled  twelve  miles,  he  estimates  that 

he  has  been  walking  at  the  rate  of  four  miles  an  hour.     We  may 

watch  him  for  one  hour  and  find  that  he  goes  four  miles.     Or  we 

may  time  him  for  any  distance  and  make  the  calculation  that  in 

one  hour  he  will  travel  four  miles.     The  relation  between  t,  the 

number  of  hours,  s,  the  miles  travelled,  and  v,  the  speed  in  miles 

per  hour,  is  evidently 


But  this  notion  no  longer  serves  us  in  the  case  of  the  falling 
stone,  which  changes  its  speed  from  moment  to  moment. 
Observe,  however,  that  the  rule 

s 

v—t 

in  no  way  depends  on  the  particular  distance  we  choose  to 
measure.  Provided  we  have  the  means  of  determining  very  short 
times  and  distances  we  may  calculate  the  speed  as  well  from 
noting  the  distance  travelled  in  the  millionth  part  of  a  second,  as 
from  that  travelled  in  a  day. 

Precisely  this  method  is  employed  in  finding  the  muzzle 
velocity  of  a  rifle  bullet,  or  cannon  ball.  Two  screens  of  tinfoil 
are  set  up,  a  short  distance  apart  in  front  of  the  rifle,  and  these 
are  made  parts  of  two  electric  circuits  through  which  currents 
pass  to  two  electromagnets.  The  magnets  hold  back  pens  that 
are  arranged  to  press  upon  a  sheet  of  paper  pasted  on  a  drum 
which  rotates  by  clockwork.  So  long  as  the  screens  are  intact, 
the  pens  describe  lines  on  the  paper;  but  when  a  screen  is  broken, 
the  corresponding  pen  instantly  flies  back  and  makes  a  nick  in  the 
line,  thus :  ' 


&       a  a  a 

a seconds  from  clock. 

b record. 

Fig.  51. 


80  MECHANICS  [CHAP. 

If  we  know  the  distance  between  the  screens,  and  the  speed 
of  the  paper  on  the  drum,  and  measure  the  lengths  aa,  ab,  we  can 
find  the  average  speed  of  the  bullet  for  the  short  distance  between 
the  screens  ;  and  by  making  this  distance  less  and  less  we  arrive 
at  the  conception  of  the  speed  of  the  bullet  at  a  definite  point  of 
its  path. 

In  the  language  of  the  Differential  Calculus  a  very  short 
distance  is  indicated  by  the  symbol  As,  and  a  very  short  time  by 

As 
A&     The  speed  at  a  given  instant  will  then  be  the  value  of  — 

in  the  limiting  case  when  both  As  and  A£  are  vanishingly  small, 
and  t  includes  the  given  instant.  This  limiting  value  is  written 

ds 

-J-,  and  the  notion  of  such   a  limit,  as  measuring  the  rate  of 

change  of  some  quantity  s  (in  our  case  a  distance)  as  compared 
with  some  other  quantity  t  (in  our  case  the  time),  is  the  funda- 
mental idea  out  of  which  the  Differential  Calculus  grew.  We 
mention  it  here  for  the  sake  of  its  importance,  though  we  shall 
not  use  it  in  our  further  calculations. 

72.     In  the  study  of  the  falling  stone  there  is  another  notion 
that  we  cannot  do  without.     Such  questions  arise 

4.    Acceleration.  HIT  n- 

as  these  :  Do  all  bodies  in  falling  get  up  speed  at 
the  same  rate  ?  Does  the  same  stone  increase  its  speed  by  the 
same  amount  in  each  second  of  its  fall  ?  We  are  thus  forced  to  - 
the  idea  of  the  rate  at  which  the  speed  changes.  The  name 
Acceleration  is  given  to  this  idea,  and  it  is  convenient  to  apply  it 
in  cases  where  the  speed  is  diminishing,  as  well  as  when  it  is 
increasing,  considering  the  acceleration  positive  when  the  speed 
increases,  and  negative  when  it  diminishes. 

Acceleration  clearly  bears  to  velocity  the  same  relation  that 
velocity  bears  to  distance  travelled.  It  may  be  measured  on  the 
same  principles.  If  it  be  found  that  in  t  seconds  a  body  has 
acquired  a  new  velocity  of  v  feet  per  second,  the  acceleration  a 
will  be  given  by 


As  before,  we  see  that  v  and  t  may  be  as  small  as  we  choose, 
provided  we  have  the  means  of  measuring  them,  without  affecting 


IX]  GALILEO   AND   THE   BEGINNINGS   OF    DYNAMICS  81 

the  principle;  and  that  a  body  may  have  an  acceleration  varying 
from  moment  to  moment  exactly  as  in  the  case  of  velocities. 

73.  Recurring  to  the  experiments  on  the  inclined  plane, 
Galileo  perceived  that  a  falling  body  had  a  constant  acceleration, 
the  same  at  all  parts  of  its  fall,  since  in  each  second  the  speed 
increases  by  equal  increments.  Since  we  know  that  the  speed 
acquired  in  one  second  is  32'2  feet  per  second,  we  can  now 
calculate  the  speed  acquired  in  any  number  of  seconds  from  the 
formula 

?;  =  32-2  x  t, 

and  the  distance  fallen  through,  from  the  formula 
s  =  ^  x£2  =  16'lx£2.       ,  . 

We  may  now  sum  up  Galileo's  investigations  of  the  motion  of 
a  falling  body  in  the  statement  that  its  weight  produces  a  constant 
acceleration  downwards,  measured  by  the  speed  gained  in  each 
second  (or  lost,  if  the  body  is  rising),  viz.  32  '2  feet  per  second. 
This  acceleration  is  usually  denoted  by  the  letter  g,  and  for  rough 
calculations  (unless  otherwise  stated)  it  may  be  taken  as 


74.  One  other  point  due  to  Galileo  must  be  mentioned. 
5.  The  Path  of  a  Since  the  stone  acquires  during  every  second  of 
projectile.  iis  faj}  an  extra  spee(j  of  ^'2  feet  per  second, 

the  action  of  the  weight  in  producing  speed  is  clearly  independent 
of  any  speed  the  body  may  previously  possess.  For  it  gains  as 
much  speed  during  the  third  second,  for  example,  when  it  starts 
the  second  with  speed  of  64*4,  as  during  the  first  second,  when  it- 
starts  from  rest. 

Galileo  at  once  extends  this  to  the  case  where  a  body  has,  to 
begin  with,  a  velocity  in  some  other  direction  than  the  vertical. 

Let  a  stone  be  thrown  horizontally  from  A  with  a  speed  v. 
Its  motion  will  consist  of  two  parts. 

(1)  As  a  heavy  body,  it  will  in  t  seconds  fall  through  a 
vertical  height  gt-j2,  and  if  this  were  all,  it  would  arrive  at  JV, 
where 


MECHANICS 


[CHAP. 


(2)     But  meanwhile  the  horizontal  velocity  will  carry  it  to  the 
right  through  a  distance  A  T  =  vt,  since  nothing  is  hindering  or 


Fig.  52. 


helping  its  horizontal  motion,  and  therefore  by  the  First  Law  of 
Motion  (§  70)  its  horizontal  speed  will  remain  unchanged. 

Galileo  sees  that  it  will  thus  arrive  at  P,  since  both  motions 
will  take  place  simultaneously.  Plotting  a  number  of  points 
such  as  P,  corresponding  to  different  times,  and  noting  that  in 
every  case 

PNZ  =  vH*  =  —  x  ?r-  =  —  x  AN. 
9       2       9 

he  finds  that  the  stone  will  describe  a  parabola. 

We  shall  deal  with  the  theory  of  projectiles  more  fully  later 
on,  and  with  another  discovery  due  to  Galileo,  which  can  only  be 
mentioned  here,  viz.  the  constancy  of  the  time  of  oscillation  of  a 
pendulum,  a  discovery  which  Galileo  was  the  first  to  apply  to 
timing  the  pulse  in  disease. 


IX]  GALILEO   AND   THE   BEGINNINGS   OF   DYNAMICS  83 

EXAMPLES. 

(0  =  32.) 

1.  From  observations  of  the  eclipses   of  Jupiter's  moons,  which  take 
place  too  early  when  the  earth  is  between  the  sun  and  Jupiter,  and  too  late 
when  the  earth  is  on  the  opposite  side  of  the  sun  to  Jupiter,  Roemer  con- 
cluded that  light  requires  16  minutes  36  seconds  to  cross  the  earth's  orbit. 
Taking  this  diameter  to  be  195,600,000  miles,  find  the  velocity  of  light  in 
miles  per  second. 

2.  The  velocity  of  sound  in  air  at  the  ordinary  temperature  is  1120  feet 
per  second.     A  thunder  clap  is  observed  to  follow  the  flash  of  lightning  at  an 
interval  of  8  seconds.     How  far  off  is  the  storm  if  the  time  required  by  the 
light  to  reach  the  observer  is  taken  as  zero  ? 

3.  Express  a  velocity  of  60  miles  an  hour  in  feet  per  second. 

4.  A  train  300  feet  long  passes  a  telegraph  post  in  12  seconds,  and, 
gaining  speed  steadily,  passes  another  post  a  quarter  of  a  mile  further  on  in 
10  seconds.     Find 

(1)  the  average  speed  at  each  post  in  miles  per  hour ; 

(2)  the  average  speed  between  posts  ; 

(3)  the  time  taken  to  travel  from  one  to  the  other  ; 
and  compare  the  acceleration  of  the  train  with  that  of  gravity. 

5.  How  long  does  it  take  a  body  to  fall  down  a  vertical  precipice  of  2000 
feet? 

6.  A  stone  is  dropped  from  a  cliff  into  the  sea,  and  is  seen  to  reach  the 
water  in  4^  seconds.     What  is  the  height  of  the  cliff  ? 

7.  A   stone  dropped   down  a  well  is  heard  to  strike   the  water  after 
4j  seconds.     The  temperature  being  just  above  freezing  point,  the  velocity 
of  sound  is  1024  feet  per  second.     What  is  the  depth  of  the  well  ? 


6—2 


CHAPTER  X. 

HUYGHENS   AND  THE   PKOBLEM   OF  UNIFORM   MOTION  IN 
A  CIRCLE.     "  CENTRIFUGAL   FORCE." 

75.  GALILEO  in  the  course  of  his  investigations  of  the  motion 
of  falling  bodies,  had  been  led  to  the  following  conclusions  : — 

(1)  The   weight   of  a   body,  if  free   to   act,  produces  in  it 
a  constant  downward  acceleration,   such   that  it  gains  speed  at 
the  rate  of  g  =  32'2  feet  per  second  in  every  second  of  its  fall. 

(2)  The  distance  fallen  through  in  a  time  t  seconds  is  gt2/2. 

(3)  If  the  weight  is  counteracted,  as  when  the  body  arrives 
upon  a  smooth  horizontal  plane,  its  speed  is  no  longer  altered,  but 
it  moves  forward  uniformly  in  a  straight  line. 

(4)  If  the   body   is  projected  with  any  speed  in  any  other 
direction,  the  action  of  the  weight  is  in  no  way  affected  by  this 
new  speed,  but  the  motion  of  the  body  is   compounded   of  the 
motion  of  a  falling  body  and  the  uniform  speed  in  a  given  direction 
initially  imparted  to  it. 

76.  The  next  case  to  be  investigated  was  that  of  a  body 
moving  round  in  a  circle  with  uniform  speed,  as  when  a  stone  is 
placed  on  a  smooth  horizontal  table,  so  as  to  neutralize  the  effect 
of  its  weight,  and  then  whirled  round  at  the  end  of  a  string.     It 
will  be  found  that,  when  a  fair  speed  has  been  attained,  the  hand 
which  holds  the  string  is  practically  motionless   at   the   centre 
of  the  circle,  but  is  conscious  of  a  steady  outward  pull  on  the 
string. 

This  problem  was  investigated  by  C.  Huyghens  (1629 — 1695) 
some  of  whose  many  brilliant  services  to  Mechanics  must  be  more 


OF  THE 

UNIVERSITY 

OF 


D.    CONSTANTINVS    'HVGENS     E,QVES 
TOPARCHA    SV  YLRCQAt 


CHAP.  X] 


CENTRIFUGAL   FORCE 


85 


fully  noticed  later  on.  Here  we  will  give  his  solution  of  this 
particular  problem  in  more  modern  form. 

Once  Galileo's  ideas  were  abroad,  it  was  natural  to  ask  :  Why 
does  not  the  stone  move  onwards  in  a  straight  line  with  unchanging 
speed  ?  The  answer  is  obvious.  It  is  subject  to  a  pull,  applied  by 
means  of  the  string,  of  the  same  nature  as  the  weight  of  the  stone, 
since  the  weight  might  be  supported  by  such  a  pull,  and  so 
balanced.  Just  as  the  weight  of  a  falling  body  gives  it  a  constant 
acceleration  downwards,  this  pull  must  give  the  stone  an  acceler- 
ation along  the  string  at  each  point  of  its  path,  by  which  it 
constantly  acquires  velocity  towards  the  centre,  and  is  deflected 
from  the  tangent  at  that  point. 

Before  finding  what  this  acceleration  is,  let  us  observe  that  a 
string  cannot  be  stretched  by  means  of  a  pull  applied  to  one  end 
only.  The  hand  applies  a  pull  inwards  at  the  centre,  and  we  are 
conscious  of  an  apparent  outward  pull  exerted  by  the  moving 
stone.  This  is  the  so-called  "  centrifugal  force."  In  studying  the 
motion  of  the  stone  it  must  be  remembered  that  the  stone  is 
subject  to  the  pull  inwards. 

77.     To  find  the  acceleration  a  P  T 

of  a   stone   describing   a   circle    of 
radius  r  with  uniform  speed  v. 

(1)  Direction.      The    accelera- 
tion must  be  directed  always  towards 
the  centre  of  the  circle.     For  since 
there   is   no   change   in   the    speed 
with  which  the  stone  moves  along 
the  circle,  any  new  velocity  acquired 
must  be  at  right  angles  to  the  circle 
at  that  point,  i.e.  along  the  radius. 

(2)  Magnitude.     Suppose  the  string  cut  when  the  stone  is  at 
P.     Then  the  stone  would   move   along   the    tangent,  with    un- 
changed speed,  and  in  a  very  short  time  t  seconds,  would  travel  a 
distance 

=  vt. 


Fig.  53. 


Again,  suppose  the  stone  to  be  at  rest  at  P,  and  then  acted  on 
by  the  pull  of  the  string  for  the  same   very  short  time   t.     By 


86  MECHANICS  [CHAP. 

Galileo's  law  of  falling  bodies,  the  acceleration  produced,  a,  would 
cause  it  to  fall  through  a  distance 


If  the  time  t  be  very  short  indeed,  say  one  billionth  of  a  second, 
the  direction  of  the  string,  and  therefore  of  the  acceleration,  will 
be  parallel  to  PO  throughout. 

Now  the  actual  motion  of  the  stone  is  compounded  of  both 
these  motions  taking  place  simultaneously  ;  and  it  arrives  at  Q. 

Since  Q  is  a  point  on  the  circle,  the  distances  PN,  PT  are  not 
independent,  but  are  connected  by  definite  geometrical  relations. 
For  instance,  if  TQ  be  produced  to  cut  the  circle  in  Q',  we  know 
that 


Substituting  the  values  of  PT,  TQ,  we  have 
a*V*x4V-HI**ft. 

When  the  time  t  is  made  very  short,  TQ'  approaches  more  and 
more  nearly  to  POP,  and  since  the  form  of  the  law  connecting 
TP,  TQ  remains  the  same,  however  small  t  is  made,  we  see  that 
when  we  arrive  at  the  actual  case  of  the  stone  changing  its  direction 
from  point  to  point, 


and 


. 
2r  .  t2      r 

This  may  be  put  in  another  form  which  is  often  convenient, 
since  we  frequently  know  the  radius  of  the  circle,  and  the  periodic 
time,  that  is  the  time  in  which  the  stone  makes  one  complete 
revolution,  so  as  to  arrive  at  the  point  from  which  it  started. 

Let  this  time  of  one  revolution  be  T  seconds. 


rnu 

Then  V 

V*        4-7T2 

a  =  r  =  "T^r- 

Since  the  radius  to  the  stone  describes  in  T  seconds  an  angle 
whose   circular  measure  is  2?r,  its  angular  velocity,  measured  in 

9 

radians  per  second,  is  -^  .     Let  this  be  denoted  by  &>.     Then  the 
acceleration  may  be  expressed  in  still  a  third  way, 

a  =  eoV. 


X]  CENTRIFUGAL   FORCE  87 


EXAMPLES. 

1.  Compare  the  "centrifugal   force"  on   a  body  at  the  equator   with 
gravity,  taking  the  earth's  radius  as  3963  miles,  and  the  time  of  revolution  to 
be  86164  seconds. 

2.  A  flywheel   10  feet  in  diameter  makes  40    revolutions  a  minute. 
Compare  the  "  centrifugal  force  "  at  the  rim  with  gravity. 

3.  A  train  runs  round  a  curve  of  radius  600  feet  at  60  miles  an  hour. 
What  acceleration  must  it  receive  inwards  ? 

4.  A  stone  is  whirled  round  in  a  vertical  plane  at  the  end  of  a  string 
2  ft.  6  ins.  long.     Shew  that  in  order  that  it  may  describe  perfect  circles,  its 
velocity  at  the  lowest  point  must  not  be  less  than  20  feet  per  second. 

5.  In  a  "  centrifugal  railway  "  the  cars,  after  descending  a  steep  incline, 
run  round  the  inside  of  a  vertical  circle  20  feet  in  diameter,  making  a  com- 
plete turn  over.     Shew  that  if  there  were  no  friction,  they  must  start  from  a 
point  not  less  than  5  feet  above  the  top  of  the  circle. 


CHAPTER  XL 

FINAL  STATEMENT   OF  THE   PEINCIPLES   OF  DYNAMICS. 

EXTENSION  TO   THE   MOTIONS  OF  THE   HEAVENLY  BODIES. 

LAW   OF   UNIVERSAL  GRAVITATION.     NEWTON. 

"Qui  genus  humanum  ingenio  superavit." 

78.  HUYGHENS'    theorems   concerning   circular  motion  were 
published  in  1673  in  his  Horologium  Oscillatorium,  which   con- 
tained   many    other    discoveries    of    capital    importance ;    both 
geometrical,  such  as  the  properties  of  cycloids,  evolutes,  and  the 
theory  of  the  circle  of  curvature  ;  and  practical,  as  the  invention 
and  construction  of  the  pendulum  clock,  the  escapement,  and  the 
method  of  determining  the  acceleration  of  gravity  by  means  of 
pendulum  observations. 

Galileo's  ideas  were  evidently  becoming  familiar.  It  was 
recognized  that  a  motion  would  continue  unchanged  unless  there 
were  some  circumstance  to  interfere  with  it,  or,  as  we  should  say, 
unless  some  force  acted  on  the  body  to  alter  its  motion  ;  that  if 
such  a  force  acted,  it  would  produce  a  change  of  speed,  or  an 
acceleration,  from  which  the  motion  could  be  calculated ;  and  that 
motion  in  a  curve  would  result  from  the  combination  of  a  deflecting 
force  with  a  motion  already  existing,  as  in  the  case  of  projectiles, 
and  motion  in  a  circle. 

79.  The  problem  that  stood  next  in  order  for  solution  was  the 
most  imposing  and  difficult  that  has  ever  been  achieved  by  the 
human  mind.     It  was — To  explain  the  movements  6f  the  heavenly 
bodies,  other  than  the  fixed  stars,  i.e.  of  the  moon,  the  planets, 
their  satellites,  and  the  comets. 


OF  THE 

UNIVERSITY 


CHAP,  xi]  KEPLER'S  LAWS  89 

The  first  steps  had  already  been  taken.  Copernicus  had  shewn 
that  the  complicated  movements  of  the  planets,  which  appeared  to 
advance,  stand  still,  recede,  and  then  advance  again  with  baffling 
irregularity,  when  viewed  by  a  spectator  on  the  earth,  could  be 
reduced  to  comparative  order  and  simplicity,  if  the  sun  were 
regarded  as  the  fixed  point  about  which  they  took  place.  The 
planets,  the  earth  being  now  one  of  them,  apparently  described 
circular  orbits  about  the  sun. 

80.  Closer  observation  shewed  that  this  was  not  exactly  the 
case.     Next,   Kepler  with   incredible  patience    had  deduced  the 
true  laws  of  their  motion  from  a  life-long  study  of  Tycho  Brahe's 
observations.     He  found  that : — 

(1)  The  planets  describe  ellipses  about  the  sun  in  one  focus. 

(2)  The  areas  swept  out  by  the  radius  vector  in  any  orbit  are 
proportional  to  the  times. 

And  he  had  found,  near  the  end  of  his  life,  a  third  law  con- 
necting the  different  orbits. 

(3)  The  squares  of  the  periodic  times  are  proportional  to  the 
cubes  of  the  semi-axes  major  (or  of  the  mean  distances). 

81.  Attempts  had  even  been  made  to  explain  the  planetary 
movements  on  mechanical  principles,  i.e.  to  trace  a  relation  between 
them  and  such  cases  of  motion  as  were  already  familiar.     Thus 
Kepler  himself  suggested  that  the  planets  were  carried  round  by 
spokes  or  radii  attached  to  the  sun ;  and  Des  Cartes  invented  a 
theoty  of  Vortices  according  to  which  each  planet  was  maintained 
in  motion  by  a  whirl  or  eddy  in  a  fluid  which  filled   all  space. 
But  these  guesses  arose  from  a  false  idea — that  motion  required 
something  to  keep  it  up  ;  and  this  was  contrary  to  Galileo's  First 
Law  of  Motion.     What  was  needed  was,  not  a  force  to  sustain  their 
motion,  but  a  deflecting  force,  that  might  cause  them  to  move 
perpetually  out  of  the  straight  line  along  the  curve  of  their  orbits. 
And  since   the    ellipses   they  described   differed  but  little   from 
circles,  and  their   speed  in  their  courses  varied  very  slightly,  it 
looked  as  if  any  disturbing  force  must  act  almost  at  right  angles 
to  their  direction  of  motion,  and  therefore  towards   the   central 
body. 


90  MECHANICS  [CHAP. 

82.  Seven  years  before  Huyghens  published  his  Horologium 
Oscillatorium,  these  ideas  had  been  clearly  grasped  by  a  young 
graduate   of  Trinity  College,   Cambridge,  who   was   destined  to 
evolve  from   them   the  complete  solution  of  the  great  problem, 
and  to  bring  every  known  motion  in  the  universe  beneath  the 
sway  of  a  single  law.     As  an  incident  in  the  course  of  his  work, 
he  completed  the  fundamental  principles  of  Dynamics,  and  stated 
the  Laws  of  Motion  in  a  form   which   with  the  aid   of  proper 
mathematical   analysis,   suffices   for    the    solution    of    all    other 
mechanical  problems. 

Isaac  Newton,  by  universal  consent  the  greatest  name  in  the 
roll-call  of  Science,  was  born  on  Christmas  day  1642  at  the  Manor 
House  of  Woolsthorpe,  a  hamlet  about  six  miles  from  Grantham 
in  Lincolnshire.  His  father  was  a  yeoman  farmer.  His  mother, 
Hannah  Ayscough,  already  widowed  when  Newton  was  born,  is 
spoken  of  as  "the  widow  Newton,  an  extraordinary  good  woman." 
One  of  her  brothers  held  a  neighbouring  living,  and  was  a 
graduate  of  Trinity  College,  Cambridge.  Small  hint  in  such 
ancestry  of  the  genius  which  has  impressed  contemporaries  and 
posterity  alike  as  almost  superhuman  ! 

Newton  was  educated  at  Grantham  Grammar  School  till  the 
age  of  fifteen,  and  shewing  no  aptitude  for  farming,  was  on  the 
advice  of  his  uncle  sent  back  to  school  and  in  1661  to  Cambridge. 
He  graduated  in  1664 ;  was  made  Fellow  of  his  College  in  1667 ; 
and  Lucasian  Professor  of  Mathematics  1669,  succeeding  Barrow 
who  had  noted  his  unparalleled  genius.  Meanwhile  as  an  under- 
graduate he  had  discovered  the  Binomial  Theorem  in  Algebra, 
and  had  begun  the  invention  of  his  method  of  Fluxions,  now 
known  as  the  Differential  Calculus.  1665  was  the  year  of  the 
Great  Plague.  The  whole  College  was  sent  down,  and  Newton 
returned  to  Woolsthorpe,  there  for  a  quiet  year  to  ponder  the  new 
ideas  he  had  gathered  at  the  University. 

83.  It  would  be  presumptuous  to  speculate  on  the  workings 
of  a  mind  like  Newton's.     Fortunately  he  has  himself  described 
the   course   of  his   thoughts,  and   there   are   other   accounts  by 
Wharton  and  by  Pemberton  based  on  his  lectures  and  conversa- 
tions in  after  years.     Thinking  over  Kepler's  Laws  in  the  light  of 


XI]  NEWTON   AND    KEPLER'S    LAWS  91 

Galileo's  dynamical  principles,  he  was  led  to  see  that  the  planets 
could  be  made  to  describe  circular  orbits  with  uniform  speed,  if 
they  were  acted  on  by  a  force  emanating  from  the  sun,  whose 
intensity  was  inversely  proportional  to  the  square  of  the  distance. 
For  since  (§  77)  a  body  describing  a  circle  of  radius  r  in  a  time 
T  has  an  acceleration  to  the  centre 


and  since  by  Kepler's  third  law  T-  for  the  different  planets  is  pro- 
portional to  r3,  the  accelerations  must  be  inversely  proportional  to 
r2.  If  the  attracting  force  were  looked  on  as  an  emanation  from 
the  sun,  one  might  almost  expect  such  a  law  of  diminution  of  its 
intensity.  For  the  areas  of  the  spheres  which  have  to  be  affected 
at  greater  and  greater  distances  increase  as  the  squares  of  their 
radii,  and  hence  the  intensity  at  any  particular  spot  on  any  sphere 
would  be  inversely  as  the  square  of  its  radius. 

84.  But  would  the  same  law  hold  for  the  actual  case,  i.e.  for 
elliptic  orbits  about  the   sun  in  one  focus?     In  the  first  place 
Kepler's  second  law  shewed  that  the  force  must  still  act  towards 
the  sun  (§§  225-6).     To  prove  that  the  law  of  the  inverse  square 
was  the  correct  law,  and  the  only  correct  law,  for  an  elliptic  orbit 
described  about  a  centre  of  force  in  one   focus  (not   about   the 
centre  of  the  ellipse,  for  that  requires  a  different  law,  §  229)  was 
a  mathematical   feat    that    required    the    genius   of  a    Newton. 
It  was,  in  fact,   the  intellectual   part   of  his   achievement,  and 
may  be  easily  appreciated  even  now  by  any  one    who,  with  a 
fair   training    in    Conic    Sections,    will    try   to   work   out    §  232 
for  himself. 

85.  It  is  not  so  easy  to  realize,  because  the  fact  has  become 
so  familiar,  the   audacity  of  imagination  by  which  Newton  dis- 
cerned, first  in  the  case  of  the  moon,  that  the  force  which  regulates 
the  courses  of  the  heavenly  bodies  is   no   more   than   ordinary 
gravity,  which  pulls  a  stone  to  the  earth.     According  to  the  well 
known  anecdote  the  great  flight  of  fancy  was  taken  when  Newton, 
convinced  of  the  need  of  such  a  central  force,  directed  towards  the 
earth,  to  account  for  the  moon's  motion,  but  unwilling  to  adopt 


92 


MECHANICS 


the  theory  till  he  could  lay  his  finger  on  the  force,  observed  the 
fall  of  an  apple  in  the  orchard  at  Woolsthorpe*.  "The  earth 
pulls  the  apple,  though  not  connected  with  it.  Why  should  it  not 
pull  the  moon  ?  " 

When  you  think  of  it,  this  is  not  so  daring  after  all.  For  the 
earth  pulls  stones  at  all  heights  accessible  to  us.  Why  should  it 
cease  to  do  so  even  at  the  height  of  the  moon  ?  We  should  even 
expect  this  according  to  the  principle  of  continuity. 

86.  But  is  it  so  ?  Newton  made  the  simple  calculation,  using 
the  data  at  his  disposal. 

The  moon's  distance  is  sixty  radii  of  the  earth.     Gravity  at 

the    moon    should    be    ^-    as  p 

powerful  as  at  the  surface  of 
the  earth,  (i.e.  one  radius  from 
the  centre).  At  the  surface  of 
the  earth  it  pulls  a  stone  through 
16£2  feet  in  t  seconds,  by  Gali- 
leo's formula;  therefore  through 
16  x  602  feet  in  a  minute.  If  it 
is  gravity  which  holds  the  moon 
in  its  orbit,  the  moon  should  fall 

through  ^  of  16  x  602,  i.e.  16 

feet  per  minute  towards  the 
earth. 

Let  PQ  be  the  distance  travelled  by  the  moon  in  one  minute. 
Then  the  distance  it  falls  towards  the  earth  is 


ON2     PQZ 
PN  =         =    r      approximately, 


where  R  is  the  earth's  radius. 

*  We  are  permitted  to  believe  the  story,  for  it  is  explicitly  stated  to  be  the  fact 
by  Conduitt,  his  assistant  at  the  Mint,  and  husband  of  his  favourite  niece  ;  by 
Voltaire,  who  had  it  from  Mrs  Conduitt  ;  and  by  E.  Greene,  on  the  authority  of 


XI]  NEWTON.      GRAVITATION  93 

27r  x  60  x  R 
PQ=       39343 
since  the  moon's  period  is  39343  minutes. 

For  the  value  of  R  Newton  had  only  the  nautical  estimate 
that   every   degree   was    60   miles,  so   that   a   circumference,  or 
ZjrR  =  60  x  360  miles. 
Thus 

2?r  x  ZirR  x  602      2  x  3'14  x  60  x  360  x  602  x  5280 

120x393432    =  120  x  39343* 

=  13-88  feet. 

This  is  too  small.  If  gravity  were  the  force  acting,  the  moon 
should  fall  through  16  feet  per  minute. 

87.  Newton's  behaviour  in  face  of  this  disappointing  result  is 
as  marvellous  an  instance  of  scientific  reserve,  as  his  daring  guess 
was  of  scientific  imagination.  At  the  age  of  twenty-three  the 
secret  of  the  universe  is  almost  within  his  grasp.  Nay,  it  is 
certain  that  Kepler's  laws  can  be  explained  by  a  central  force 
inversely  as  the  square  of  the  distance.  But  "  hypotheses  non 
tingo."  Rather  than  base  his  theory  on  anything  but  a  "vera 
causa,"  a  force  known  to  exist  on  other  grounds,  he  lays  aside  the 
whole  subject  in  silence. 

Six  years  later,  in  1672,  Picard  of  Paris  communicated  to  the 
Royal  Society  a  new  and  careful  determination  of  the  size  of  a 
degree,  making  it  69£  miles  instead  of  60.  When  this  came  to 
Newton's  knowledge,  he  took  out  his  old  papers,  and  made  the 
calculation  with  the  new  numbers,  and  it  is  said  that  as  it  became 
clear  that  the  result  would  accord  with  theory,  his  excitement  was 
so  great  that  he  could  not  see  the  paper  to  finish  his  work*. 

For  two  years  he  now  threw  himself  into  the  labour  of  working 
out  the  detailed  application  of  his  discovery  with  such  ardour  and 
concentration  of  mind  that  he  often  forgot  to  take  foodf. 

Martin  Folkes,  who  was  associated  with  Newton  as  Vice-President  of  the  Boyal 
Society  when  Newton  was  President.  The  tree  from  which,  according  to 
tradition,  the  apple  fell,  was  blown  down  in  1820,  and  some  of  its  wood  has'  been 
preserved. 

*  There  seems  to  be  no  authority  for  this  particular  story,  first  given  by  Kobison 
in  1804. 

t  There  are  many  anecdotes  illustrating  Newton's  absentmindedness  during 
these  years. 


94  MECHANICS  [CHAP. 

88.  A  great  deal  had  to  be  done.     First  a  new  method  had 
to  be  devised,  for  treating  things  which  were  almost  insensibly 
but  yet  continually  varying,  such  as  the  direction  of  motion  in 
the  curved  path  of  a  planet,  and  its  gradually  changing  speed ; 
and  the  varying  force  which  acted  on  it  as  it  moved  closer  to  the 
sun  or  farther  away  from  it.     From  the  doctrine  of  limiting  ratios 
of  vanishingly  small  quantities  employed  by  Newton  arose   the 
Method  of  Fluxions  and  afterwards  the  Differential  Calculus. 

89.  Next,  the  principles  of  Mechanics  had  to  be  collected  and 
completed,  and  put  in  a  shape  convenient  for  the  calculation  of 
orbits  described  under  any  laws  of  force,  and  particularly  that 
found  in  nature,  the  inverse  square  of  the  distance,  about  which  a 
number  of  special  propositions  had  to  be  proved.     Then  these  had 
to  be  verified  for  the  planets  and  their  satellites,  and  the  comets 
also  brought  under  the  law. 

But  if  the  earth,  the  sun,  and  the  planets  attract  other  bodies 
according  to  this  law,  why  not  also  much  smaller  bodies  such  as 
meteorites  ?  Again,  must  we  not  suppose  that  every  part  of  the 
earth  shares  in  producing  the  attraction  ?  Nay,  must  not  every 
part  attract  every  other  part  ?  Or,  rather,  must  not  every  particle 
in  the  universe  attract  every  other  particle  according  to  the  law 
of  the  inverse  square  of  the  distance  ? 

90.  Before  it  was  possible  to  accept  this  idea,  Newton  had. 
to  shew  how  a  sphere,  such  as  the  earth,  composed  of  particles 
attracting  according  to  the  inverse  square  of  the  distance,  would 
act  on  an  external  body.    He  worked  out  a  number  of  propositions 
shewing  how  a  sphere  would  act  on  a  particle  inside  it,  on  its 
surface,  and  at  any  distance  outside.     For  instance,  in  the  last 
case  the  attraction  is  the  same  as  if  the  whole  substance  were 
collected  at  the  centre  of  the  sphere. 

91.  Another   application  was  based  on   Kepler's  third   law. 
Knowing  the  distance  of  any  planet  from  the  sun  and  the  length 
of  its  year,  and  also  the  length   of  the  lunar  month  and  the 
distance  of  the  moon  from  the  earth,  Newton  was  able  to  compare 
the  mass,  or  "  quantity  of  matter "  in  the  sun  with  that  of  the 
earth. 


XI]  NEWTON.      THE   MOON'S   MOTION  95 

92.  But  perhaps  his  most  amazing  achievement  was  his 
treatment  of  the  perturbations  of  the  moon's  orbit.  The  moon 
is  attracted  by  the  sun  as  well  as  by  the  earth,  and  hence  her 
motion  varies  in  a  number  of  ways  from  a  true  elliptic  motion 
about  the  earth.  The  most  important  of  these  "  inequalities  "  or 
irregularities  were  worked  out  by  geometrical  methods  which  no 
one  has  been  able  to  advance  beyond  the  point  where  Newton 
left  them. 

These  are : 

(a)  The  evection,  a  periodical  change  in  the  eccentricity  of 
the  ellipse,  discovered  by  Hipparchus  and  Ptolemy. 

(b)  The  variation,  by  which  new  and  full  moon  occur  a  little 
too  early,  and  the  quadratures  a  little  too  late;  and 

(c)  The  annual  equation,  a  variation  in  the  other  perturba- 
tions depending  on  the  varying  position  of  the  earth  in  her  orbit. 

These  two  were  discovered  by  Tycho  Brahe. 

(d)  The  regression  of  the  nodes ;  and 

(e)  The  variation  of  the  inclination  of  the  moon's  orbit. 

These  were  at  the  time  being  observed  by  Flamsteed  at 
Greenwich. 

(/)  The  progression  of  the  apses,  whereby  the  moon's  orbit 
turns  round  in  its  own  plane  through  3°  in  a  year. 

Two  other  inequalities : 

(g)     The  inequality  of  the  apogee, 

(h)     The  inequality  of  the  nodes, 

were  predicted   by  Newton,  never  having  been  noticed  by  the 
observers  before. 

Newton's  calculation  of  (/)  gave  only  1J°,  one  half  the 
observed  amount.  D'Alembert,  Clairaut,  and  others  of  the  great 
analytical  mathematicians  attempted  to  account  for  this  curious 
discrepancy,  but  arrived  at  the  same  result ;  till  at  last  Clairaut 
found  that  a  number  of  terms  had  been  omitted  in  the  series 
as  unimportant,  which  turned  out  to  be  not  negligible,  and  when 
these  were  included,  the  result  was  correct.  It  was  not  till 
Professor  Adams,  one  of  the  discoverers  of  Neptune,  was  editing 


96  MECHANICS  [CHAP. 

the  Newton  papers  in  the  possession  of  the  Earl  of  Portsmouth, 
that  MSS.  were  discovered  shewing  that  Newton  had  himself 
reworked  the  calculations,  and  found  out  the  cause  of  error, 
but  had  not  published  the  correction ! 

93.  Newton  made  several  other  remarkable   applications  of 
his  theory. 

From  the  time  of  revolution  of  the  earth,  considered  as  a 
fluid  mass,  he  calculated  its  oblateness,  and  conversely,  from  the 
observed  shape  of  Jupiter  he  estimated  the  length  of  Jupiter's  day. 

Then  from  the  shape  of  the  earth  so  deduced,  combining  its 
attraction  with  the  effect  of  the  "  centrifugal  force  "  of  its  rotation, 
he  compared  the  force  of  gravity  at  the  poles  and  the  equator. 

Again,  from  the  attractions  of  the  sun  and  moon  on  the  earth's 
equatorial  protuberance,  he  explained  and  calculated  the  precession 
of  the  equinoxes. 

Finally,  he  worked  out  the  theory  of  the  Tides  from  the 
unequal  attraction  of  the  moon  upon  the  solid  nucleus  of  the 
earth,  and  on  the  nearer  and  farther  parts  of  the  ocean,  taking 
account  of  special  conformations  of  land  and  water  in  different 
parts  of  the  earth ;  explained  the  spring  and  neap  tides  as  the 
resultant  of  the  tides  due  to  the  moon  and  tbe  sun ;  reckoned  the 
height  of  the  solar  tide  from  his  known  mass ;  and  then  from  the 
observations  of  the  spring  and  neap  tides  deduced  a  first  estimate 
of  the  mass  of  the  moon. 

94.  "In  1683,  among  the  leading  lights  of  the  Royal  Society, 
the  same  sort  of  notions  about  gravity  arid  the  solar  system  began 
independently  to  be  bruited.     The  theory  of  gravitation  seemed 
to  be  in  the  air,  and  Wren,  Hooke,  and  Halley  had  many  a  talk 
about  it." 

"Hooke  shewed  an  experiment  with  a  pendulum,  which  he 
likened  to  a  planet  going  round  the  sun.  The  analogy  is  more 
superficial  than  real.  It  does  not  obey  Kepler's  laws ;  still  it  was 
a  striking  experiment.  They  had  guessed  at  a  law  of  inverse 
squares,  and  their  difficulty  was  to  prove  what  curve  a  body 
subject  to  it  would  describe.  They  knew  it  ought  to  be  an  ellipse 
if  it  was  to  serve  to  explain  the  planetary  motion,  and  Hooke  said 


XT]  NEWTON.      GRAVITATION  97 

he  could  prove  that  an  ellipse  it  was ;  but  he  was  nothing  of  a 
mathematician,  and  the  others  scarcely  believed  him.  Undoubt- 
edly he  had  shrewd  inklings  of  the  truth,  though  his  guesses  were 
based  on  little  else  than  a  most  sagacious  intuition.  He  surmised 
also  that  gravity  was  the  force  concerned,  and  asserted  that  the 
path  of  an  ordinary  projectile  was  an  ellipse,  like  the  path  of  a 
planet — which  is  quite  right*." 

In  January  1684  Wren  offered  a  prize — a  book  worth  forty 
shillings — to  Hooke  and  Halley  if  either  of  them  could  produce  a 
proof  that  a  body  under  the  law  of  inverse  squares  would  describe 
an  ellipse.  But  as  nothing  was  forthcoming,  in  the  following 
August  Halley  made  a  journey  to  Cambridge  and  put  the  question 
to  Newton,  who  then  for  the  first  time  told  him  that  he  had  worked 
it  all  out  a  dozen  years  before  !  Halley  communicated  his  discovery 
to  the  Royal  Society,  and  at  their  request  Newton  revised  and 
completed  his  papers  and  allowed  them  to  be  published.  The 
Principia  appeared  in  1687,  and  in  connection  with  this 
momentous  event  it  should  not  be  forgotten  that  not  only  was  its 
publication  brought  about  by  Halley,  but  that  he  saw  the  work 
through  the  press  and  defrayed  the  cost  at  his  own  risk. 

I 

95.  It  is  obvious  that  the  conception  of  universal  gravitation 
introduces  a  wonderful  simplicity  and  order  into  our  views  of  the 
varied  and  complicated  motions  of  the  heavenly  bodies.  It  sets  us 
at  a  new  point  of  view  from  which  the  intricate  movements  which 
had  puzzled  the  ages  are  seen  to  fall  into  their  places  as  parts  of  a 
general  scheme  whose  secret  can  be  expressed  in  a  single,  simple, 
universally  applicable  statement. 

But  Newton  is  very  careful  to  warn  us  that  he  has  not 
discovered  the  causes  of  these  movements;  not  even  framed  a 
hypothesis  to  account  for  the  attraction  of  gravitation.  He  has 
found  a  formula  which  describes  with  extreme  simplicity  and 
universality  how  the  motions  go  on.  So  that  by  means  of  it  we  can 
comprehend  them  in  all  their  bewildering  complexity,  or  com- 
municate our  knowledge  to  others,  or  calculate  how  they  will  go 
on  happening,  and  use  our  predictions  to  shape  our  conduct 
wisely.  Not  even  the  fall  of  a  stone  is  explained,  in  the  sense  that 

*  Lodge,  Pioneers  of  Science. 
C.  7 


98  MECHANICS  [CHAP. 

a  cause  is  found  for  it.  Nothing  is  explained  in  this  sense,  but 
by  the  law  of  gravitation  the  most  complicated  motions  known  to 
us  in  the  universe  are  brought  into  relation  with  the  simple  case 
of  the  stone,  and  what  was  remote  and  strange  is  reduced  to  what 
is  familiar  enough. 

96.  We  have  dwelt  so  long  on  the  discovery  of  the  law  of 
gravitation  partly  because  of  its  intrinsic  interest  and  importance, 
and  partly  for  the  sake  of  the  light  it  throws  on  the  method  by 
which  science  advances.     But  we  must  now  turn  to — what  more 
immediately   concerns   us — several   very  important  steps  in  the 
development  of  Mechanical  theory  to  which  Newton  was  Ted  in 
the  course  of  solving  his  great  problem. 

97.  Newton  realized  more  clearly  than  had  ever  been  done 

before  that  motion  could  be  altered  not  only  by 

1.  Generalization  i  -n        •  i  •    i 

of  the  idea  of  means  of  pushes  and  pulls,  in  which  we  are 

conscious  of  the  muscular  effort  by  which  we  effect 
the  change,  but  by  other  circumstances,  such  as  the  supposed 
attraction  of  the  earth,  and  the  known  attractions  of  electrified  and 
magnetized  bodies,  which  Dr  Gilbert  of  Colchester  had  recently 
written  about  so  admirably. 

Now  when  we  make  a  muscular  effort,  we  say  that  we  exert  a 
Force.  Newton  generalized  this  idea  so  as  to  make  it  include  all 
the  other  cases,  and  gave  it  the  definition  still  current. 

Definition  of  Force.  Force  is  any  circumstance  which  changes 
or  tends  to  change  a  body's  state  of  rest  or  of  uniform  motion  in  a 
straight  line. 

2.  The  Parallel-  98.     This  principle,  already  dimly  grasped  by 
ogram  oi  Forces.       Galileo  and  Steviuus,  was  now  stated  explicitly. 

99.     In  view  of  the  law  of  gravitation  it  appeared  that  a  body 

3.  The  concept        might  have  very  different  weights  according  to  its 

position  with  regard  to  the  earth,  and  indeed  would 
have  no  weight  at  all,  if  it  were  placed  at  the  centre  of  the  earth, 
or  at  a  very  great  distance  from  any  attracting  body.  Nevertheless 
the  object  remains  unchanged  even  when  its  weight  disappears. 


XI]  WEIGHT   AND    MASS  99 

We  cannot  remove  objects  to  the  centre  of  the  earth  or  to 
immense  distances.  But  let  us  neutralize  the  weight  of  some 
object.  Thus  a  curling  stone  on  smooth  ice  has  its  weight 
supported  by  the  upward  pressure  of  the  ice.  But  a  considerable 
effort  is  required  to  set  it  in  motion,  or  to  stop  it  when  once 
started. 

Hang  two  equal  weights  by  a  string  over  a  smoothly  running 
pulley.  They  will  rest  in  any  position.  But  an  effort  is  required 
to  get  them  into  motion,  and  this  effort  will  be  greater  the 
greater  the  size  of  the  weights,  and  greater  for  leaden  weights 
than  for  weights  of  equal  size  made  of  brass. 

Again,  a  heavy  fly-wheel  on  a  smooth  axle  will  rest  in  any 
position,  but  requires  effort  to  start  it  or  stop  it. 

From  a  consideration  of  such  cases  it  is  clear  that  there  is 
something  about  a  body,  not  its  weight,  which  has  a  great  effect  in 
determining  its  behaviour  when  acted  on  by  forces,  and  which, 
so  far  as  we  know,  remains  unchanged  even  when  the  weight  is 
neutralized  or  altered.  The  term  "  Inertia "  was  introduced  to 
indicate  that  bodies  had  no  power  to  produce  changes  in  their  own 
motion,  and  offered  an  apparent  resistance  to  changes  of  motion, 
which  had  to  be  overcome  by  an  effort  of  some  kind  from 
outside. 

There  thus  emerges  a  very  important  distinction  between  the 
weight  of  a  body,  which  is  variable  and  depends  on  its  position 
with  regard  to  some  other  attracting  body,  such  as  the  earth ;  and 
something  else,  apparently  an  unchangeable  attribute  of  the  body, 
which  determines  how  it  will  respond  to  the  action  of  forces 
tending  to  change  its  motion. 

Newton  rather  unfortunately  called  this  "the  quantity  of 
matter"  in  the  body.  The  modern  term  is  "mass."  We  shall 
define  this  term  more  precisely  later,  but  at  present  call  the 
attention  of  the  student  to  the  distinction  between  mass  and 
weight,  which  Newton  was  the  first  to  realize. 

100.     To  clear  the  way  for  the  solution  of  his  great  problem 
4.  The  Laws     Newton  began  by  laying  down  three  '  Axioms '  or 
of  Motion.          Lawg  of  Motion  :— 

I.     Every  body  continues  in  its  state  of  rest  or  of  uniform 

7—2 


100  MECHANICS  [CHAP.  XI 

motion  in  a  straight  line  except  in  so  far  as  it  is  compelled  to 
change  that  state  by  impressed  force. 

II.  Change  of  Motion  is  proportional  to  the  impressed  force 
and  takes  place  in  the  direction  of  the  force. 

III.  To  every  action  there  is  an  equal  and  opposite  reaction. 

These  Laws,  or  at  all  events  the  first  two  of  them,  are  little 
more  than  the  first  explicit  statement  of  ideas  which  were  already 
generally,  though  vaguely,  held.  But  it  was  an  important  step 
to  have  them  clearly  stated  in  a  form  which,  in  spite  of  many 
criticisms,  still  holds  the  field. 

The  First  is  Galileo's  *  Principle  of  Inertia/  that  a  body  has  no 
power  in  itself  to  change  its  own  state  of  rest  or  motion. 

The  Second  is  the  fundamental  Law  of  Dynamics.  Its  full 
meaning  and  all  that  it  implies  will  be  discussed  in  Book  II. 

The  Third  was  perhaps  the  only  absolutely  new  point  in 
Newton's  statement.  So  soon  as  it  became  necessary  to  calculate 
the  movements  of  two  bodies  each  of  which  attracted  the  other, 
the  question  arose  "  What  is  the  relation  between  the  two  mutual 
attractions  ? "  Newton's  answer  was  that  the  two  attractions 
would  be  equal ;  and  he  generalized  the  statement  for  forces  of 
all  kinds  in  his  Third  Law  : — 

"  To  every  Action  there  is  an  equal  and  opposite  Reaction." 

These  laws  have  proved  sufficient  for  the  solution  of  all 
problems  in  Dynamics.  All  that  has  happened  since  has  been  a. 
mere  matter  of  mathematical  development  of  their  consequences. 

With  the  Laws  of  Motion  the  historical  evolution  of  the 
fundamental  principles  employed  in  Mechanics  may  be  considered 
completed. 


BOOK   II. 

MATHEMATICAL    STATEMENT    OF  THE 
PEINCIPLES. 


INTRODUCTION. 

101.  WE  have  seen  in  Book  I.  how  the  fundamental  principles 
of  Mechanics  were  gradually  won  from  experience  by  men  of 
genius  engaged  in  attempting  to  solve  problems  that  either  forced 
themselves  on  their  attention  by  the  practical  importance  of  their 
consequences,  as  in  the  case  of  the  Simple  Machines,  or  attracted 
them  by  their  own  impressive  grandeur  and  their  bearing  on 
questions  of  philosophy. 

Some  of  these  principles  can  be  mutually  deduced  from  each 
other.  They  are  equally  valid  as  results  of  experience,  and  would 
serve  equally  well  as  the  starting  point  of  the  subject.  The 
particular  order  in  which  they  arose  was  largely  a  matter  of 
historical  accident. 

We  shall  now  give  a  more  connected  and  precise  statement  of 
the  subject,  selecting  that  order  which,  after  Newton,  makes  the 
Laws  of  Motion  the  starting  point  in  experience  of  all  the  rest. 

And  as  we  are  to  study  motion  as  produced  by  force  in  real 
bodies,  let  us  begin  by  clearing  up  our  ideas  about  motion  by  itself 
apart  from  any  consideration  of  what  is  moving  or  why  it  moves. 
This  power  of  abstraction,  or  attending  to  one  thing  at  a  time,  is 
of  great  value  in  science. 

Let  a  small  mirror  be  held  in  the  sun  so  that  a  spot  of  light  is 
reflected  on  to  the  walls,  ceiling,  and  furniture  of  a  room.  Even  a 
perfectly  regular  turning  of  the  mirror  will  set  the  spot  moving 
with  baffling  variations  both  of  speed  and  direction.  When  it 
passes  from  wall  to  ceiling,  there  is  an  instantaneous  change  of 
direction  such  as  never  happens  to  a  heavy  body,  no  matter  what 
the  force  applied  to  it.  We  could  study  the  position,  velocity,  and 


104  MECHANICS 

change  of  velocity  of  such  a  spot,  without  any  reference  to  the 
laws  of  motion  according  to  which  motion  is  found  to  arise  and 
change  in  real  bodies. 

This  part  of  the  subject,  which  deals  with  pure  motion  in  the 
abstract,  is  called  Kinematics. 

The  part  dealing  with  the  motion  of  real  bodies  under  the 
action  of  forces  is  called  Kinetics. 

The  two  branches  together  are  often  called  Dynamics.  The 
special  cases  where  the  forces  concerned  happen  to  be  in  equi- 
librium are  usually  classed  together  under  the  title  Statics. 


CHAPTER  XII. 

KINEMATICS. 

102.  CLERK  MAXWELL  says : 

"  The  most  important  step  in  the  progress  of  every  science 

is  the  measurement  of  quantities.      Those  whose 

curiosity  is  satisfied  with  observing  what  happens 

have  occasionally  done  service  by  directing  the  attention  of  others 

to  the  phenomena  they  have  seen;  but  it  is  to  those  who  endeavour 

to  find  out  how  much  there  is  of  anything  that  we  owe  all  the 

great  advances  in  our  knowledge." 

In  order  to  measure,  or  express  the  exact  amount  of  any 
quantity,  two  things  are  required :  (1)  a  unit,  or  standard  quantity 
of  the  same  nature  as  that  to  be  measured ;  and  (2)  a  number  to 
indicate  how  many  such  units  or  parts  of  a  unit  are  contained  in 
the  given  quantity.  Thus  a  sum  of  money  may  be  expressed  as 
20  shillings,  or  4*86  dollars.  But  it  would  be  impossible  to  convey 
to  a  stranger  any  idea  of  a  sum  of  money  unless  we  had  previously 
come  to  an  understanding  as  to  the  purchasing  power  of  some 
definite  amount,  such  as  a  shilling  or  a  dollar,  and  could  both  of 
us  count. 

The  number  required  to  express  any  given  amount  will 
evidently  be  greater  the  smaller  the  unit  we  employ,  and  vice 
versd.  The  measure  of  a  quantity  is  inversely  proportional  to 
the  unit  in  which  it  is  expressed. 

103.  All  the  civilised  governments  have  united  in  establishing 

an  International  Bureau  of  Weights  and  Measures 
in  the  Pavilion  de  Breteuil,  in  the  Pare  of  St  Cloud 

at  Sevres,  near  Paris.     Here  are  kept  the  standards  of  length  and 

mass. 


106  MECHANICS  [CHAP. 

The  unit  of  length  is  the  International  Metre,  which  is  defined 
as  the  distance,  at  the  melting  point  of  ice,  between  the  centres  of 
two  lines  engraved  upon  the  polished  surface  of  a  platiniridium 
bar,  of  a  nearly  X-shaped  section,  called  the  International  Prototype 
Metre.  The  international  metre  is  authoritatively  declared  to  be 
identical  with  the  former  French  metre,  or  metre  des  archives. 
This  was  intended  to  be  one-ten-millionth  part  of  a  quadrant  of  a 
terrestrial  meridian.  .  But  as  the  value  of  a  quadrant  came  to  be 
more  accurately  determined,  and  moreover  is  changing,  the  actual 
bar  constructed  has  been  made  the  standard,  and  succeeding 
determinations  of  the  quadrant  are  now  expressed  in  terms 
of  it. 

The  accepted  unit  of  length  in  all  scientific  works  except  those 
of  British  engineers,  is  the  centimetre,  or  one-hundredth  part  of 
the  standard  metre. 

The  British  unit  of  length  is  the  Imperial  Yard  which  is  the 
distance  at  62°  F.  between  the  centres  of  two  lines  engraved  on. 
gold  plugs  inserted  in  a  bronze  bar  usually  kept  walled  up  in  the 
Houses  of  Parliament  at  Westminster. 

The  foot,  or  third  part  of  the  standard  yard,  is  often  employed 
as  the  unit  in  British  works. 

For  measuring  great  distances  multiples  of  these  units  are 
used,  such  as  the  kilometre  and  the  mile ;  very  small  lengths  are 

A  DECIMETRE  DIVIDED  INTO  CENTIMETRES  AND  MILLIMETRES. 


Jffl 


1 

2 

3 

INCHES  AND  TENTHS. 

Fig.  55.     1  metre  =  39-37079  British  inches. 

often  expressed  in  submultiples  such  as  the  micron,  or  one- 
thousandth  part  of  a  millimetre,  i.e.  the  millionth  part  of  a 
metre. 

The  relation  between  the  two  units  is  shewn  in  Fio\  55. 


XII]  KINEMATICS  107 

104.  The  universal  unit  of  time  is  the  mean  solar  day  or  its 
Time  one  86400th  part,  which  is  called  a  second. 

This  is  the  time  during  which  the  earth  turns 
on  its  axis  through  a  certain  small  angle  with  reference  to  the 
fixed  stars. 

Any  time  is  measured  by  the  number  of  seconds  it  contains. 

It  will  be  seen  that  in  measuring  the  speed  of  any  body  by  the 
number  of  units  of  length  it  travels  over  in  a  unit  of  time,  we  are 
really  comparing  its  motion  with  another  motion,  viz.  that  of  the 
earth  on  its  axis,  just  as  we  compare  a  length  with  another  length. 
Should  there  be  any  change  in  the  standard  length  or  in  the  rate 
of  the  earth's  rotation  on  its  axis,  our  measures  would  lose  their 
meaning.  Clerk  Maxwell  suggests  that  those  authors  who  think 
their  works  likely  to  outlast  the  present  condition  of  the  earth, 
would  do  well  to  express  their  lengths  in  terms  of  the  wave-length 
of  some  particular  ray  in  the  spectrum,  and  times  in  terms  of  the 
periodic  time  of  vibration  of  such  a  ray,  quantities  which  we  have 
every  reason  to  believe  will  remain  constant  so  long  as  the  physical 
universe  retains  its  identity. 

At  the  request  of  the  French  Government,  Professor  Michelson 
has  determined  the  value  of  the  standard  metre  in  wave-lengths 
of  the  red,  green,  and  blue  rays  of  cadmium,  and  finds  that 

1  metre  =  1,553,163-5  wave-lengths red      ray 

=  1,966,249-7  „  green    „ 

=  2,083,372-1  „  blue      „ 

at  15°  C.  and  760  mm.  pressure. 

105.  The  position  of  a  point  in  a  straight  line  is  fixed  when 

we  know  its  distance  from  some  fixed  point  of 

Position. 

reference,  or  origin,  in  the  straight  line,  and  the 
direction  in  which  this  distance  is  to  be  measured. 

It  is  convenient  to  prefix  the  positive  sign  to  all  distances 
measured  in  one  direction,  e.g.  towards  the  right  if  the  line  is 
horizontal,  and  the  negative  sign  to  all  distances  measured  in  the 
opposite  direction  (towards  the  left  if  the  line  is  horizontal),  from 
their  respective  starting  points.  A  distance  will  thus  be  positive, 
even  though  it  lies  entirely  to  the  left  of  the  origin,  provided  it 
is  measured  towards  the  right. 


108 


MECHANICS 


[CHAP. 


With  this  convention,  the  algebraic  sum  of  all  the  distances 
(with  their  proper  signs)  travelled  by  a  point  starting  from  the 
origin  will  always  give  us  its  position,  and  the  sign  of  the  sum 
will  tell  us  whether  it  is  to  the  right  or  left  of  the  origin. 


Fig.  56. 

Thus  if  a  man  starts  from  a  town  0,  and  walks  3  miles  due 
east,  his  position  at  P  is  indicated  by  +  3.  If  he  now  walk  5  miles 
more  to  the  east,  he  will  arrive  at  Q,  where  OQ  =  +  3  +  5  =  +  8. 

Let  him  now  walk  6  miles  westward.  He  will  then  be  found 
at  R  where 

OR  =  +  3  +  5-6  =  +  2. 

Finally  let  him  continue  westward  for  5  miles.  He  will  be  at 
St  where 


106.  The  position  of  a  point  in  a  plane  may  be  fixed  by 
Cartesian  Co-ordinates,  so  called  after  their  inventor  Des  Cartes, 
or  by  Polar  Co-ordinates. 

(1)     Cartesian  Co-ordinates. 

Y 


M 


N          X 


Y' 

Tig.  57.    Cartesian  Co-ordinates. 


XII]  KINEMATICS  109 

Two  lines  of  reference,  XOX' ',  YOY',  are  chosen,  usually  at 
right  angles.  These  are  called  the  axes.  Through  any  point  P 
parallels  to  the  axes  are  drawn,*  cutting  off  lengths  ON,  OM. 
These  are  the  co-ordinates  of  the  point  P.  Lengths  along  YOY'  are 
counted  positive  if  measured  upwards,  and  negative  if  measured 
downwards,  from  their  starting  points,  whether  the  lengths  them- 
selves be  above  or  below  0.  When  ON,  OM  are  known,  the 
position  of  P  is  fixed  by  lines  drawn  through  N  and  M  parallel  to 
the  axes. 

(2)     Polar  Co-ordinates. 

The  position  of  P  may  also  be  fixed  by  the  distance  OP  and 
the  angle  POX  —  0  through  which  OP  must  revolve  from  OX  to 
reach  P. 

Angles  turned  through  in  the  opposite  direction  to  that  of  the 
hands  of  a  clock  (counter-clockwise)  are  counted  positive.  Those 
turned  through  in  the  clockwise  direction  are  negative. 

Since  the  point  P  is  considered  as  carried  through  the  angle  6 
by  the  revolving  radius  OP,  OP  is  called  the  radius  vector.  The 
radius  vector  and  the  angle  0  are  called  the  polar  co-ordinates  of  P 
with  reference  to  the  pole  0. 

Two  systems  are  employed  for  measuring  angles  in  works  on 
Trigonometry.  The  unit  angle  in  one  of  them  is  the  right  angle, 
with  its  subdivisions  into  degrees,  minutes,  and  seconds. 

In  the  other,  the  system  of  Circular  Measure,  the  unit  angle  is 
the  radian,  i.e.  the  angle  subtended  at  the  centre  of  any  circle  by 
an  arc  equal  to  its  radius.  This  is  an  angle  of  about  57°  19'. 


Fig.  58.    Polar  Co-ordinates. 

This  system  is  often  convenient  in  Mechanics,  especially  for 
dealing  with  rotations,  for  the  length  s  of  the  arc  of  a  circle  of 


110  MECHANICS 

radius  r  described  by  P  when  the  radius  vector  revolves  through 
an  angle  whose  circular  measure  is  0,  is  given  by 

s  =  r0. 

107.  Definition.     A  point  is  said  to  move  when  it  changes  its 

position  with  reference  to  surrounding1  objects,  or 

Motion.  •       i        .  i« 

some  particular  object  chosen  for  reference.  The 
only  motions  known  to  us  are  thus  relative  motions.  But  this  is 
no  restriction,  since  in  practic^  we  are  only  interested  in  relative 
motions.  We  want  to  know  for  instance  how  to  avoid  a  collision 
with  another  ship,  or  to  strike  a  fort  with  a  shell,  and  for  all  such 
purposes  a  knowledge  of  relative  motion  suffices. 

For  most  purposes  motions  are  referred  to  the  surface  of  the 
earth,  in  spite  of  its  own  rapid  and  complicated  motion.  In 
astronomy  the  sun  is  chosen  for  reference  so  long  as  we  confine 
ourselves  to  the  solar  system.  The  motion  of  the  sun  itself  is 
referred  to  the  general  body  of  so-called  fixed  stars. 

• 

108.  Definition.     The  velocity  of  a  point  is  the  rate  at  which 

it  is  changing  its  place. 

Velocity.  3    .&  r. 

The  unit  velocity  is  that  of  a  point  which 
passes  over  a  unit  length  in  a  unit  time.  It  is  usually  therefore 
a  velocity  of  one  foot  or  else  one  centimetre  per  second. 

Any  other  velocity  is  measured  by  the  number  of  units  of 
velocity  it  contains.  This  is  the  same  as  the  number  of  units  of 
length  passed  over  in  a  unit  of  time,  if  the  velocity  be  uniform. 
This  may  be  determined  (§  71)  from  the  distance  travelled  in  a 
very  short  interval  of  time,  without  waiting  for  a  whole  second. 

There  is  no  difficulty  in  extending  this  notion  to  the  more 
usual  case  where  the  velocity  is  variable,  i.e.  is  changing  from 
moment  to  moment.  Let  the  interval  of  time  considered  be  chosen 
smaller  and  smaller,  but  always  so  as  to  include  the  particular 
instant  at  which  the  velocity  is  to  be  estimated.  The  numbers 
representing  the  distance  travelled  and  the  time  occupied  in 
traversing  it  may  thus  be  made  vanishingly  small,  but  their  ratio, 
which  measures  the  velocity,  has  a  finite  value  however  small  they 
are  made.  Common  experience  has  familiarized  the  idea  of  speed 
at  a  particular  moment,  independently  of  the  length  of  time  for 


XII]  KINEMATICS  111 

which  it  is  maintained.  Thus  a  train  may  pass  a  particular  signal 
post  at  sixty  miles  an  hour,  and  stop  a  few  hundred  yards  beyond 
it.  Everyone  understands  what  is  meant  by  the  statement,  and 
knows  that  if  the  speed  had  been  maintained  unaltered,  the  train 
would  have  covered  sixty  miles  in  the  next  hour. 

109.  Since  a  velocity  is  specified  when  we  know  its  magnitude 

and   its   direction,  it   may  be   represented  bv  a 

Geometrical  repre-  ..,.  _.  «        ».  .", 

sentation  of  straight  line.     For  the  line  may  be  drawn  in  the 

given  direction,  and  of  such  a  length  as  to  re- 
present the  magnitude  on  any  convenient  scale.  It  is  convenient 
to  use  the  term  "  speed  "  to  indicate  the  magnitude  of  a  velocity 
irrespective  of  its  direction,  velocity  implying  that  the  direction 
also  must  be  taken  account  of. 

110.  Definition.     The  acceleration  of  a  point  is  the  rate  at 

which  it  is  changing  its  velocity. 

Acceleration. 

The  unit  acceleration  is  that  of  a  point  which 
gains  one  unit  of  velocity  in  one  unit  of  time.  It  is  usually  there- 
fore an  acceleration  of  one  foot  per  second  in  a  second,  or  of  one 
centimetre  per  second  in  a  second. 

Any  other  acceleration  is  measured  by  the  number  of  units  of 
acceleration  it  contains.  This  is  the  same  as  the  number  of  units 
of  velocity  gained  in  a  unit  of  time,  if  the  acceleration  be  uniform. 
This  may  be  determined  from  the  velocity  gained  in  a  very  short 
interval  of  time,  without  waiting  for  a  whole  second. 

The  notion  may  be  extended  to  variable  accelerations,  exactly 
as  in  the  case  of  velocities,  by  considering  the  velocities  gained  in 
a  vanishingly  short  interval  of  time. 

Acceleration  thus  stands  to  velocity  as  velocity  stands  to 
distance  travelled,  and,  like  velocity,  may  be  represented  by  a 
properly  drawn  straight  line. 

111.  We  require  formulae  expressing  the  connection  between 
The  kinematic      tne  acceleration  a,  the  velocity  v  at  any  time,  the 
formulae.  distance  travelled  s,  and  the  time  elapsed  t. 

(1)     Uniform  Velocity. 

In  this  case   there  is  but   one   formula,  but   care   must   be 


112  MECHANICS  [CHAP. 

taken  on  no  account  to  use  it  in  questions  involving  variable 
velocity. 

If  a  point  move  for  t  seconds  with  uniform  speed  vt  the  distance 

travelled  s  is  given  by 

s  =  vt, 
with  its  equivalents, 

s  s 

v=- ;     t  =  -. 

t  v 

112.     (2)     Uniform  Acceleration. 

Let  a  be  the  acceleration.  Then  if  the  point  start  from  rest,  it 
will  acquire  a  units  of  velocity  in  every  second,  and  at  the  end  of 
t  seconds  it  will  have  a  velocity  v,  where 

v  =  at  (1). 

To  find  the  distance  travelled. 

The  velocity  at  the  middle  moment  (not  the  middle  of  the  path), 

i.e.  after  t/2  seconds  have  elapsed,  is  -^  . 

Compare  the  actual  motion  of  the  point  with  the  motion  of  a 
point  which  starts  at  the  same  instant  with  the  velocity  ^ ,  and 

maintains  its  speed  unchanged  throughout.  For  every  moment 
in  the  first  half  of  the  time  when  the  first  point  is  moving  more 
slowly  than  the  second,  there  is  a  corresponding  moment  in  the 
second  half  when  it  is  moving  just  as  much  faster.  So  that  in  the 
end  the  two  points  will  cover  the  same  ground. 
Therefore 

s  =  (average  velocity)  x  (time  of  motion) 
at 
2"  X  * 

=  i«*2    (2). 

Another  form  may  be  given  to  this  result,  which  is  important 
when  we  wish  to  connect  the  velocity  acquired  directly  with  the 
space  passed  over. 
As  above, 

s  =  (average  velocity)  x  (time  of  motion) 

v  v  ,  . 

=  5  x  (since  v  =  at) 

&  a 


XII]  KINEMATICS  113 

•y2 
We  write  this  ^  =  as  ......................  .  .......  (3). 


113.  If  the  point,  instead  of  starting  from  rest,  has  an  initial 
velocity  u,  these  formulae  admit  of  a  simple  modification.  The 
effect  of  the  acceleration  in  producing  either  new  velocity,  or  extra 
distance  travelled,  or  extra  half-square  of  the  velocity,  has  simply 
to  be  added  to  what  is  due  to  the  initial  velocity. 

Thus,  corresponding  to 

(Formulae  for  point  initially  at  rest)  (Formulae,  initial  vel.  u) 

v  =  at,  we  have  v  =  u  +  at, 


Otherwise  thus  : 

From  the  definition  of  acceleration 

v  —  u 

a  =  —  -  —  ,        .  *.  v  =  u  +  at. 
t 

Again,  the  average  velocity  in  this  case  is 


Now        s  =  (average  velocity)  x  (time  of  motion) 

-!)    «     - 

— 

Also         s  =  (average  velocity)  x  (time  of  motion) 
v + u  v—u 

—  _  x\  ~ 


c. 


114  MECHANICS  [CHAP. 


EXAMPLES. 

1.  Remembering  that  a  velocity  of  60  miles  an  hour  is  equivalent  to 
88  feet  per  second,  write  down  in  feet  per  second  velocities  of  15,  20,  36  miles 
an  hour;  and  in  miles  an  hour  velocities  of  8,  11,  and  40  feet  per  second. 

\J2.  It  takes  light  3'315  years  to  come  from  the  star  a  Centauri  to  tho 
earth.  If  the  velocity  of  light  is  186,000  miles  per  second,  and  the  radius  of 
the  earth's  orbit  is  92,370,000  miles,  express  the  distance  of  a  Centauri  in 
radii  of  the  earth's  orbit. 

Y>3.  Taking  the  earth's  orbit  to  be  circular,  find  the  mean  velocity  of  the 
earth  in  its  orbit. 

\^  4.  Find  the  velocity  of  a  point  at  the  equator  due  to  the  rotation  of  the 
earth  on  its  axis,  if  the  earth's  radius  is  3963  miles. 

5.  A  bullet  is  fired  through  two  screens  1  metre  apart,  and  the  interval 
required  to  pass  from  one  to  the  other,  as  recorded  on  a  chronograph,  is 
•0036  second.  Express  its  velocity  in  centimetres  and  feet  per  second. 

\^  6.  A  steamer  approaching  a  coast  with  vertical  cliffs  in  a  fog  whistles, 
and  the  echo,  as  timed  by  a  stop-watch,  is  heard  after  8|  seconds.  One 
minute  afterwards  she  whistles  again  and  the  echo  is  heard  after  4f  seconds. 
How  far  is  she  then  off  shore?  How  fast  is  she  going?  How  soon  will  she 
strike  if  she  goes  on  ?  (Sound  travels  a  mile  in  5  seconds.) 

7.  An  enemy's  guns  are  heard  4f  seconds  after  the  flash.  Express  the 
range  in  yards,  assuming  the  velocity  of  sound  to  be  1120  feet  per  second. 

Vs^8.  A  military  band  marches  off  at  the  rate  of  9  steps  in  5  seconds, 
covering  2  feet  6  inches  every  step.  When  they  have  made  122  steps  after 
passing  a  spectator,  they  appear  to  him  to  be  exactly  out  of  step  with  the 
music.  What  was  the  velocity  of  sound  that  day  ? 

9.  A  train  moves  from  rest  and  after  one  minute  has  a  velocity  of  30 
miles  an  hour.     What  is  its  acceleration  ? 

10.  A  body  starts  with  velocity  30  and  after  8  seconds  has  velocity  90. 
What  is  its  acceleration  ? 

11.  A  body  has  acceleration  32  and  starts  with  velocity  80.     What  is  the 
velocity  after  1,  4,  and  10  seconds? 

12.  A  stone  dropped  from  a  stationary  balloon  reaches  the  ground  in 
24  seconds.     What  was  its  velocity  at  the  ground? 


XII]  KINEMATICS  115 

13.  A  stone  is  thrown  vertically  upwards  with  a  velocity  of  120  feet  per 
second,  the  acceleration  being  32  downwards.  How  soon  will  it  be  stationary  ? 
Find  its  velocity  after  3,  4,  and  7  seconds. 

\J  14.     What  is  the  acceleration  of  a  train  whose  speed  increases  from  20  to 
30  miles  an  hour  in  100  yards? 

15.  A  train  running  40  miles  an  hour  has  the  brakes  put  on  and  reduces 
its  speed  to  20  miles  an  hour  in  220  yards.     What  is  the  acceleration  ?     How 
much  farther  will  it  run  before  it  stops? 

16.  (a)  A  bullet  acquires  a  speed  of  1600  feet  per  second  while  traversing 
a  rifle  barrel  4  feet  long.     Find  the  average  acceleration. 

(6)   The  muzzle  velocity  of  a  revolver  bullet  is  600  feet  per  second, 
and  the  barrel  is  8  inches  long.     Find  the  average  acceleration. 

17.  A  point  moves  12  feet  in  1  second  and  18  feet  the  next.     How  long 
has  it  been  moving  with    uniform    acceleration    from  rest?     What  is  its 
acceleration  ?     How  far  will  it  go  in  the  next  10  seconds,  and  when  will  its 
velocity  be  81  feet  per  second? 

18.  A  body  has  initial  velocity  u  and  acceleration  a.     Find  a  formula  for 
the  space  passed  over  in  the  nth  second. 

19.  How  long  must  a  body  travel  with  the  acceleration  of  gravity  before 
it  acquires  the  velocity  of  light  ?     How  far  would  it  move  in  the  time  ? 

20.  A  bullet  is  fired  vertically  upwards  with  a  velocity  of  1600  feet  per 
second.     After  how  many  seconds  will  it  return  to  the  earth?     What  is  the 
greatest  height  reached  ? 

21.  How  far  must  a  body  fall  to  acquire  a  speed  of  400  metres  per  second  ? 

22.  A  velocity  of  15  foot-second  units  is  changed  into  5  units  while  the 
body  travels  50  feet.     What  is  the  acceleration  ?    What  would  it  have  been  if 
the  velocity  had  been  changed  to   -5  in  the  same  distance?     How  much 
longer  must  the  acceleration  have  acted  in  the  second  case,  and  how  much 
farther  will  the  body  have  travelled? 


8—2 


•CHAPTER  XIII. 

KINETICS  OF  A  PARTICLE   MOVING  IN  A  STRAIGHT   LINE. 
THE  LAWS  OF   MOTION. 

114.  IN  dealing  with  the  movements  of  real  bodies  about  us 
produced  by  our  own  muscular  efforts  we  know  from  experience 
that  the  effect  produced  by  a  given  effort  will  depend  largely  on 
the  body  to  which  it  is  applied. 

So  long  as  we  are  concerned  with  portions  of  the  same 
substance,  the  size  of  the  body  determines  the  result,  at  all  events 
approximately.  Thus  if  a  certain  effort  is  required  to  project  a 
stone  with  a  certain  speed,  something  like  twice  the  effort  will  be 
needed  to  project  in  the  same  manner  a  stone  of  twice  the  size. 

But  when  we  compare  the  effects  of  the  same  effort  on  two 
bodies  of  different  nature,  such  as  cork  and  lead,  something  else 
besides  mere  volume  has  to  be  considered. 

Newton  gave  to  this  "  something  "  which  determines  the  effect 
of  an  effort  upon  a  body,  the  name  "  quantity  of  matter  "  in  the 
body ;  rather  unfortunately,  as  has  been  said,  because  the  definition 
seems  to  raise  the  question  "  what  is  matter  ?"  a  question  which  has 
occupied  philosophers  from  the  earliest  times  without  yet  receiving 
a  generally  accepted  answer.  Newton  himself  immediately  has  to 
define  how  the  "  quantity  of  matter  in  a  body  "  is  to  be  estimated. 
He  says  that  it  is  to  be  taken  as  the  "  product  of  the  volume  of 
the  body  and  its  density."  But  as  the  only  way  of  determining 
the  density  is  first  to  find  the  quantity  of  matter  in  a  unit  of 
volume,  it  is  clear  that  we  are  landed  in  a  logical  circle. 

Fortunately  the  physicist  is  not  bound  to  enter  on  the  thorny 
paths  of  philosophy,  at  least  at  this  point.  He  may  content 


CHAP.  XIIl]  MASS   AND   FORCE  117 

himself  with  the  answer  of  the  Oxford  undergraduate,  who,  when 
asked  "What  is  mind  ?"  replied  "No  matter";  "What  is  matter?" 
"  Never  mind." 

And  yet  we  cannot  study  scientifically  the  movements  of  real 
bodies  without  being  able  to  measure  this  quantity  which  deter- 
mines the  motion  that  will  be  produced  in  them  by  a  given  effort. 
This  implies  that  we  shall  be  able  to  choose  a  unit  quantity,  and 
count  the  number  of  such  units  contained  in  a  body,  whether  of 
cork,  or  lead,  or  any  other  substance.  How  is  this  to  be  done  ?  The 
mere  volume,  we  have  seen,  will  not  help  us  to  deal  with  different 
vsubstances. 

For  shortness,  and  to  avoid  the  misleading  associations  of  the 
word  matter,  let  us  call  the  quantity,  unchangeable  so  far  as  we 
know,  which  determines  the  effect  of  a  given  effort  in  producing 
speed  in  a  body,  the  Mass  of  the  body. 

And  as  Newton  extended  the  ideas  connected  with  our 
muscular  efforts  to  all  other  cases  where  motion  is  changed  by  any 
means  whatever,  let  us  adopt  his  definition  of  Force. 

Force  is  anything  which  changes  or  tends  to  change  a  body  a 
state  of  rest  or  of  uniform  motion  in  a  straight  line. 

It  might  be  supposed  that  though  we  cannot  compare  masses 
by  comparing  their  volumes,  yet  we  might  do  so  by  comparing 
their  weights.  And  this,  as  Newton  points  out,  happens  to  be 
true  enough,  at  all  events  for  comparisons  made  at  the  same  place. 
But  we  must  not  assume  that  the  weight  of  a  body,  which  varies 
from  place  to  place,  and  would  be  nothing  at  all  at  the  centre  of 
the  earth,  and  only  one-sixth  as  great  at  the  surface  of  the  moon, 
is  a  safe  guide  in  measuring  its  mass,  which  no  physical  circum- 
stances known  to  us  will  suffice  to  change  in  the  slightest  degree. 
We  must  find  some  other  test  of  the  equality  of  two  masses. 

If  our  object  were  to  study  the  chemical  properties  of  bodies, 
we  might  legitimately  define  equal  quantities  of  two  different 
substances  as  those  which  could  neutralize  the  same  amount  of 
some  standard  reagent,  such  as  sulphuric  acid,  if  we  could  find  one 
which  acted  on  all  substances;  and  the  science  could  be  logically 
built  on  such  a  definition. 

Our  actual  purpose  in  Mechanics  is  to  study  the  effect  of  forces 
in  producing  motion.  The  proper  test  of  the  equality  of  two  masses 


118  MECHANICS  [CHAP. 

is  therefore  to  observe  whether  the  same  force  produces  the  same 
mechanical  effect  on  them.  This  indeed  is  the  test  we  instinctively 
apply  in  practice.  If,  of  two  equal  casks  lying  on  a  wharf,  one  is 
known  to  be  full  and  the  other  empty,  and  we  wish  to  find  out 
which  is  which,  we  give  each  of  them  a  kick  or  push,  and  the  one 
which  resists  us  most  is  the  full  one. 

Definition.  Two  masses  are  equal  if  the  same  force  acting  on 
each  of  them  for  the  same  time  produces  in  each  the  same 
velocity. 

To  be  sure  that  we  are  applying  the  same  force  we  might 
apply  it  by  means  of  a  spiral  spring,  taking  care  to  pull  it  so  that 
its  extension  remains  the  same  throughout.  If  we  may  assume 
that  the  physical  properties  of  the  spring  remain  unchanged,  then 
the  force  will  be  constant.  Though  this  is  not  a  practical  form  of 
experiment,  it  is  theoretically  sufficient,  and  this  is  all  that  is 
necessary  for  our  present  purpose,  which  is  to  conceive  a  test  by 
which  other  masses  may  be  set  off  equal  to  the  standard. 

115.     Two  such  units  or  standards  are.  employed. 

(1)  The  International  Kilogramme,  which  is  the  mass  of  a 
certain  cylinder  of  platiniridium  kept  at  Sevres,  and  intended  to 
be  identical  with  the  former  French  kilogramme  des  Archives. 

In  science  it  is  generally  the  Gramme,  or  thousandth  part  of 
the  kilogramme,  which  is  taken  to 'be  the  unit.  The  gramme 
was  intended  to  be  the  mass  of  a  cubic  centimetre  of  water  at  its 
temperature  of  maximum  density  3'93°  C. 

The  system  of  units  which  is  based  on  the  centimetre,  gramme, 
and  second,  as  units  of  length,  mass,  and  time  respectively,  is  called 
the  C.G.s.  system. 

(2)  The  Pound,  which  is  the   mass  of  a  certain   platinum 
weight,  called  the  British  Imperial  Pound. 

According  to  Miller's  determination 

1  pound  =  0'4535926525  kilogramme, 
1  kilogramme  =  2-204621 249  pounds. 

The  mass  of  any  other  body  is  expressed  by  the  number  of  such 
units  of  mass  (pounds  or  kilogrammes)  it  contains. 


XIIl]  COMPARISON    OF   MASSES  119 

116.  It  is  so  important  to  keep  clearly  in  mind  the  dis- 
tinction between  the  mass  and  the  weight  of  a  body,  and  also  the 
method  of  testing  the  equality  of  two  masses,  that  we  will  consider 
an  example  in  detail. 

Suppose  we  wish  to  purchase  a  pound  of  sugar.  Note  that 
from  Newton's  point  of  view  (mass  =  quantity  of  matter)  we 
should  be  glad  to  have  the  mass  as  large  as  possible ;  whereas 
the  weight,  i.e.  the  pull  with  which  it  tends  towards  the  earth,  is 
purely  an  inconvenience  when  it  comes  to  carrying  the  sugar 
home,  and  we  might  be  glad  if  it  could  be  done  away  with, 
provided  that  the  mass  were  not  thereby  diminished. 

To  determine  what  is  a  pound  of  sugar,  the  shopman  might 
set  upon  a  long  counter  two  little  wheeled  cars  exactly  alike  in 
all  respects.  In  one  of  these,  according  to  our  test,  he  should 
place  a  standard  pound  ;  in  the  other  a  quantity  of  sugar.  They 
should  then  be  successively  drawn  along  by  means  of  a  spring 
balance,  care  being  taken  that  the  reading  of  the  balance  always 
remained  the  same,  and  their  progress  timed.  If  the  sugar  were 
found  to  out-run  the  standard  pound,  more  should  be  added ;  if  it 
fell  behind,  some  taken  away,  until  at  last  both  cars  were  found  to 
gain  speed  at  the  same  rate.  We  should  then  have  exactly  one 
pound  of  sugar. 

A  simpler  way  would  be  first  to  test  two  spring  balances  by 
locking  them  together,  and  observing  whether  their  readings  were 
equal  when  they  were  pulled  apart.  Then,  drawing  the  two  cars 
along  simultaneously,  one  by  each  balance,  we  could  observe  which 
of  them  required  the  greater  force  to  gain  approximately  the  same 
speed,  and  adjust  the  amount  of  sugar  till  the  balances  gave  the 
same  reading  during  the  experiment. 

Observe  that  the  pound  is  not  a  force  but  a  mass.  The  word 
is  often  employed  to  denote  the  force  with  which  a  pound  tends 
downwards,  i.e.  the  weight  of  a  pound.  When  we  specify  a  force 
by  the  number  of  pounds  it  would  sustain,  it  is  better  always  to 
use  the  correct  but  more  cumbrous  form,  and  speak  of  a  force  of 
so  many  pound- weights,  except  when  there  can  be  no  possible 
danger  of  confusion. 

Having  defined  Force  and  Mass  and  seen  how  masses  may 
be  measured,  we  are  ready  to  study  the  laws  of  motion  as  stated 
by  Newton. 


120  MECHANICS  [CHAP. 

117.  Newton's  Laws  of  Motion. 

Law  I.  Every  body  continues  in  its  state  of  rest  or  of 
uniform  motion  in  a  straight  line,  except  in  so  far  as  it  is 
compelled  to  change  that  state  by  impressed  force. 

This  is  merely  Galileo's  principle  of  Inertia,  by  which  is  meant 
that  a  body  has  no  power  in  itself  of  altering  its  own  state  of 
motion,  whatever  that  may  be,  but  can  only  change  it  in  response 
to  some  force  applied  from  outside.  As  we  have  seen  (§  70), 
Galileo  arrived  at  the  law  from  the  principle  of  continuity  applied 
to  the  motion  of  a  body  down  an  inclined  plane.  It  is  distinctly 
contrary  to  the  views  generally  held  in  his  time,  and  by  unobservant 
people  to  this  day. 

118.  The   full    establishment   of   a   principle    of    this    kind 
generally  consists  of  four  stages : 

I.  Observation] 

TT     -r,         .        ,  \.     Induction  of  the  Law  from  facts. 

II.  Jiixperiment  j 

III.  Deduction  of  Consequences  of  the  Law. 

IV.  Verification,  by  comparison  of  the  consequences  deduced 
with  further  observations  of  facts. 

The  first  of  these  stages  is  the  most  difficult,  requires  the 
greatest  originality.  To  descry  a  new  meaning  in  a  fact  whose 
very  familiarity  blinds  our  eyes  to  its  significance,  like  Columbus- 
with  the  drift-weeds  on  the  western  shore ;  to  break  away  from 
inveterate  prejudice  to  new  points  of  view,  like  Copernicus;  to 
catch  in  a  flash  of  intuition  the  resemblance  between  remote  facts, 
as  when  Bradley  in  his  moving  boat  saw  that  the  slant  in  the  rain 
and  the  infinitesimal  shift  in  the  place  of  the  fixed  stars  were  akin, 
and  so  unravelled  the  aberration  of  light ;  to  divine  in  the  falling 
apple  the  secret  of  the  heavens ;  this  is  the  work  of  genius,  of  the 
poet's  imagination,  vivid  in  observation,  fertile  in  surmise.  Before 
a  problem  guesses  come  to  most  men,  but  in  what  surpassing 
measure  to  a  Newton ! 

I.  Once  the  guess  is  made,  it  must  be  tested.  Sometimes,  as 
in  Astronomy,  we  can  do  no  more  than  wait  till  the  event 'can 
be  observed  again,  as  with  the  phenomena  connected  with  total 
eclipses  of  the  sun,  and  transits  of  Venus. 


XIII]  THE    FIRST   LAW   OF   MOTION  121 

II.  Or  we  may  have  the  facts  under  our  control,  and  be  able 
to  repeat  them  at  pleasure,  varying  the  circumstances.     Then  we 
experiment,  seeking  to  disentangle  what  is  essential  from  what  is 
indifferent,  according  to  the  canons  laid  down  in  works  on  logic. 

These  two  stages,  Observation  and  Experiment,  constitute  the 
Induction  of  the  law  from  the  facts. 

III.  In  deducing  the  consequences  which  should  follow,  if  the 
law  be  true,  the  instrument  most  generally  employed  in  Physics 
is  Mathematics,  which  is  only  a  systematic  method  of  applying 
common  sense  with  ease  and  accuracy. 

IV.  Finally  the  results  of  calculation  are  carefully  compared 
with  fresh  observations.     It  not  unfrequently  happens  that  the 
theory  leads  to  recondite  consequences  that  would  hardly  have 
been  stumbled  on  without  its  aid.     Thus  Fresnel's  Undulatory 
Theory  of  Light  enabled  Sir  W.  R.  Hamilton  to  predict  the  conical 
refraction   in   crystals,   afterwards    observed   by   Thomas   Young 
working  from  his  directions ;  and  Adams  and  Leverrier  discovered 
Neptune  from  a  consideration  of  the  disturbances  in  the  orbit  of 
Uranus.     Such  startling  and  dramatic  verifications  lead  to  the 
rapid  adoption  of  a  principle,  but  its  final  acceptance  depends  on 
patient  comparison  of  calculation  with  observation  resulting  in 
universal  agreement  in  detail. 

119.  The  First  Law  of  Motion  cannot  be  observed  directly, 
because  we  cannot  screen  a  body  from  the  action  of  all  forces 
and  watch  its  behaviour.  Nevertheless  the  stage  of  observation 
was  fulfilled  when  Galileo  divined  it  from  motion  on  an  inclined 
plane. 

For  experiment,  the  more  we  do  to  remove  retarding  forces, 
the  longer  motion  continues.  Push  a  table  along  a  rough  floor. 
It  comes  to  rest  (through  friction)  the  moment  we  cease  pushing. 
Place  its  legs  on  castors,  and  it  will  -run  a  few  inches  after  we  let 
go.  Set  the  wheels  on  rails,  as  in  a  railway  truck,  and  once 
started  it  will  travel  a  considerable  distance.  A  block  of  ice 
thrown  along  a  sheet  of  ice  travels  a  very  long  way.  Two  equal 
weights  suspended  by  a  fine  thread  over  a  very  lightly  running 
pulley  (Atwood's  Machine,  Fig.  61  a)  balance  each  other.  But  if 


122  MECHANICS  [CHAP. 

set  in  motion,  the  system  travels  with  almost  uniform  velocity  for 
a  long  time. 

The  inventions  of  the  Perpetual  Motion  seekers  are  often  good 
instances  of  approximation  to  the  case  of  the  First  Law. 

The  stages  of  Deduction  and  Verification  for  all  the  Laws  of 
Motion  find  a  superb  illustration  in  the  Nautical  Almanac,  This 
volume  of  600  pages,  published  four  years  in  advance,  contains  on 
every  page  many  hundreds  of  predictions  of  the  places  of  the  sun, 
the  planets,  the  satellites  of  the  planets,  and  of  the  moon  among 
the  fixed  stars ;  and  the  dates,  durations,  place  of  commencement, 
path,  and  conclusion  of  eclipses,  worked  out  to  a  degree  of  accuracy 
within  the  limits  of  error  of  the  most  sensitive  modern  instruments 
of  precision.  Every  calculation  is  founded  on  the  three  Laws  of 
Motion,  applied  to  the  averages  of  long  series  of  corrected  previous 
observations.  Yet  such  is  our  confidence  in  their  truth,  that  every 
ship  captain  unhesitatingly  stakes  his  vessel  on  the  results  deduced 
from  them ;  and  it  is  safe  to  say  that  if  an  astronomer,  provided 
with  the  finest  instrument  in  the  world,  observed  even  a  minute 
departure  from  its  calculated  place  in  one  of  the  heavenly  bodies, 
it  would  never  occur  to  him  to  doubt  the  laws  of  motion,  but  he 
would  search  for  some  unusual  source  of  error  in  his  instrument, 
or  suspect  a  new  and  undetected  cause  of  disturbance,  as  did  Adams 
and  Leverrier  in  the  case  of  Neptune. 

120.  The  First  Law  states  that  unless  some  force  acts  on  a 
body  from  without,  its  motion  continues  unchanged. 

The  Second  Law  tells  us  how  the  motion  will  be  changed  when 
a  force  acts  on  the  body. 

Law  II.  Change  of  motion  is  proportional  to  the  impressed 
force,  and  takes  place  in  the  direction  of  the  force. 

By  motion  Newton  does  not  mean  velocity  only,  since  the  same 
force  will  produce  very  different  changes  of  velocity  in  different 
masses.  In  measuring  the  quantity  of  motion  we  must  there- 
fore take  account  of  the  mass  moved  as  well  as  of  the  speed 
produced. 

We  choose  for  unit  quantity  of  motion  the  quantity  of  motion 
contained  in  one  unit  of  mass  moving  with  unit  velocity.  On  the 


XIIl]  THE   SECOND  LAW   OF   MOTION  123 

British  system  this  will  be  the  quantity  of  motion  possessed  by 
one  pound  moving  with  a  speed  of  one  foot  per  second. 

M  pounds  moving  one  foot  per  second  will  contain  M  times 
as  much,  and  if  the  M  pounds  are  moving  V  feet  per  second, 
there  will  be  V  times  as  much  again.  So  that  M  pounds  moving 
V  feet  per  second  contain  a  quantity  of  motion  represented  by  MV 
such  units. 

It  is  time  to  have  a  single  name  for  this  recurrent  phrase 
q uantity  of . motion. 

Definition.  The  quantity  of  motion  in  a  body  is  called  its 
Momentum.  It  is  measured  by  the  product  of  the  mass  of  the 
body  into  its  velocity.  No  special  name  has  been  given  to  the 
unit  of  momentum. 

The  word  proportional,  in  Law  II,  is  to  be  taken  in  its  strict 
mathematical  sense ;  i.e.  questions  on  the  Second  Law  are  to  be 
worked  out  by  Rule  of  Three,  or  Proportion. 

In  measuring  the  impressed  force  we  must  take  account  not 
only  of  the  magnitude  of  the  force,  but  also  of  the  time  during 
which  it  acts;  since  the  longer  a  force  acts  the  greater  is  the 
change  of  motion  it  produces. 

The  total  effect  of  a  force  in  producing  change  of  motion  is 
called  its  Impulse  (i.e.  total  push). 

We  choose  as  unit  impulse  the  effect  of  unit  force  acting  for 
unit  time,  i.e.  for  one  second. 

P  units  of  force  acting  for  one  second  produce  P  units  of 
impulse ;  and  if  they  continue  acting  for  t  seconds,  there  will  be 
Pt  units. 

121.     We  can  now  state  the  Second  Law  as  follows : 
Momentum  produced  is  proportional    to   the  Impulse  of  the 
Force  acting,  and  is  in  the  direction  of  the  force. 
In  algebraical  symbols : 

MVvPt. 

The  sign  of  variation,  x  ,  is  inconvenient  in  this  equation, 
and  may  be  got  rid  of  by  a  proper  choice  of  units.  Now  we  have 
already  chosen  the  units  of  mass,  velocity,  and  time.  But  nothing 
lias  been  said  about  the  unit  force.  It  is  therefore  open  to  us  to 
choose  a  unit  force,  and  we  define  it  thus. 


124  MECHANICS  [CHAP. 

Definition.  The  unit  force  is  that  force  which  acting  on  unit 
mass  for  unit  time  produces  in  it  the  unit  velocity. 

Let  us  calculate,  by  rule  of  proportion  according  to  the  Second 
Law,  the  velocity  that  will  be  produced  when  P  units  of  force  act 
for  t  seconds  on  M  units  of  mass. 

By  definition  : 

1  unit  of  force  acting  on  1  unit  of  mass  for  1  second  produces 
1  unit  of  velocity. 

P  units  of  force  acting  on  1  unit  of  mass  for  1  second  produce 
P  units  of  velocity. 

P  units  of  force  acting  on  M  units  of  mass  for  1  second  produce 
~Tf  units  of  velocity. 

P  units  of  force  acting  on  M  units  of  mass  for  t  seconds  produce 
-jrj  units  of  velocity. 

.    V-Pt 
~  M' 

Provided  that  we  choose  our  unit  of  force  as  above,  therefore, 
we  may  write 


and  from  this  formula  we  can  calculate  either  the  velocity 
produced  in  a  given  mass  by  a  given  force  acting  for  a  given 
time: 

V=Pt/M  ...........................  (1), 

or  the  force  required  to  produce  a  given  velocity  in  a  given  mass 
in  a  given  time  : 

P  =  MV/t  ...........................  (2). 

122.  A  very  important  particular  case  of  (1)  is  that  in  which 
the  force  acts  for  one  second  ;    for  the  velocity  gained  in    one 
second  measures  the  acceleration  produced  in  the  mass  M  by  the 
force  P. 

Thus  a  =  P/M,  P  =  Ma. 

123.  The  simplest  way  of  solving  dynamical  problems  is  to 
use  this  formula  in  conjunction  with  the  Kinematical  formulae  of 
the  last  chapter.     Thus  : 


XIII] 


THE   UNIT   OF    FORCE 


125 


Dynamical  Formula 

Kinematical  Formulae 

From  rest 

Initial  velocity  u 

v  =  at 

v=u+at 

P 

_at2 

s  =  ut  +  ^- 

j-« 

V2      U2 

—  =  h  as 

In  general,  either  the  forces  acting  and  the  mass  acted  on  are 
given,  and  it  is  required  to  find  something  about  the  speed  gained 
or  the  distance  run  in  a  certain  time  ;  or  else  some  relation 
between  time,  speed,  and  distance  is  given,  and  it  is  required 
to  find  either  the  force  acting  or  the  mass  moved,  one  of  the  two 
being  known.  In  the  former  case  we  find  a  from  the  dynamical 

p 
formula  a=  -^  ,  and  then  use  its  value  in  the  kinematical  formulae. 

In  the  latter  we  begin  by  finding  a  from  the  kinematical  formulae, 

p 
and  then  find  P  or  M,  whichever  is  unknown,  from  a  =      . 


124.  The  unit  of  Force  has  been  defined,  but  so  far  we  have 
no  means  of  comparing  it  with  forces  more  familiar  to  us,  such  as 
the  weight  of  a  pound  or  of  a  gramme.  This  can  only  be  effected 
by  an  experiment. 

(1)     Unit  force  on  the  British  system  of  units. 

We  want  to  know  what  force  acting  on  one  pound  for  one 
second  will  give  it  a  speed  of  one  foot  per  second. 

Let  a  standard  pound  be  dropped  from  a  height,  so  that  it  is 
acted  on  solely  by  its  own  weight.  At  Greenwich  it  is  found  to 
acquire  in  one  second  a  speed  of  32*2  feet  per  second.  (§  69.) 
The  weight  of  one  pound  must  therefore  be  32'2  units  of  force 
such  as  we  have  chosen  ;  and  our  unit  is  the  32'2th  part  of  the 
weight  of  one  pound  at  the  place  where  the  experiment  is  tried. 

This  unit  is  called  the  Poundal,  or  the  British  Absolute  unit 
of  Force,  because  its  value  is  the  same  wherever  the  experiment 


126  MECHANICS  [CHAP. 

to  determine  it  is  tried.  For  if  it  be  found  that  at  some  other 
place  the  speed  acquired  in  one  second  of  fall  is  different,  for 
instance  3216  feet  per  second,  then  the  weight  of  a  pound,  as 
measured  by  a  spring  balance,  will  also  be  found  to  be  less  at  this 
place  in  the  proportion  3216  to  32'20,  so  that  the  3216th  part  of 
it  leads  to  the  same  value  of  the  unit. 

This  value  is  about  a  half-ounce  weight ;  and  whenever  the 

p 
formulae  a  —  ^-,  or  MV—Pt  are  employed,  forces,  when  given, 

must  be  expressed  in  poundals  (by  multiplying  pound-weights  by 
32*2);  and,  when  found,  will  come  out  in  poundals,  and  can  be 
converted  to  pound-weights  by  dividing  by  32*2. 

(2)     The  c.G.S.  system. 

A  gramme  acted  on  by  its  own  weight  is  found  at  Paris  to 
acquire  a  speed  of  981  centimetres  per  second,  in  one  second. 

The  absolute  unit  of  force  on  the  C.G.S.  system  is  therefore  the 
-gsjth  part  of  the  weight  of  a  gramme  at  Paris.  This  unit  is  called 
a  Dyne  (Svva/j,i,s  =  force).  It  is  not  far  from  the  weight  of  a 
milligramme.  All  forces  occurring  in  dynamical  formulae  must 
be  expressed  in  dynes  by  multiplying  gramme-weights  by  981 ; 
and  answers  expressed  in  dynes  can  be  converted  back  to  gramme- 
weights  by  dividing  by  981. 

125.     Weight  is  proportional  to  Mass. 

Galileo  was  led  to  disbelieve  the  common  opinion  of  his  time 
that  heavy  bodies  fall  more  quickly  than  light  ones.  To  settle  the 
question  he  tried  an  experiment  at  the  celebrated  Leaning  Tower 
of  Pisa.  Boxes  of  the  same  size  and  shape,  but  filled  with  different 
materials,  were  dropped  from  the  summit,  and,  as  he  expected, 
were  found  to  reach  the  earth  at  about  the  same  moment,  whether 
their  contents  were  light  or  heavy.  The  slight  outstanding 
differences  he  rightly  referred  to  the  resistance  of  the  air,  which 
has  a  greater  proportional  effect  on  the  light  bodies  than  on  the 
heavy  ones. 

Experiment.  The  student  should  verify  this  fact.  Let  an 
iron  ball,  and  a  wooden  ball  with  an  iron  nail  in  it  be  supported 
by  two  small  electromagnets  attached  to  a  wooden  bar  and  drawn 
up  to  any  height.  If  the  same  current  be  made  to  pass  round 


XIII]  WEIGHT   PROPORTIONAL   TO   MASS  127 

both  magnets,  the  balls  can  be  released  simultaneously  by 
breaking  the  circuit.  The  balls  will  reach  the  floor  almost 
exactly  together. 

Newton  repeated  this  experiment  in  a  striking  form  by 
dropping  a  guinea  and  a  feather  at  the  same  moment  in  a  long 
glass  tube  from  which  the  air  had  been  exhausted.  The  feather 
fell  like  the  metal. 

A  very  important  conclusion  follows  from  this  experiment. 
Since  all  bodies  fall  equally  fast  in  vacuo,  the  acceleration  must 

p 

be  the  same  for  each  at  every  moment  of  the  fall.     Thus  -^  is 

Jf 

the  same  for  all  bodies  at  the  same  place.  But  the  only  force 
acting  on  a  falling  body  is  its  own  weight.  Therefore  W/M  is 
the  same  for  all  bodies,  i.e.  the  weight  of  a  body  is  proportional 
to  its  mass. 

We  see  now  that  masses  may  be  compared  by  comparing  their 
weights,  a  far  more  convenient  method  than  that  of  §  116. 

Since  the  acceleration  produced  by  the  weight  of  a  body  is  the 
same  for  all  bodies,  i.e.  g  =  32'2,  or  981,  according  to  the  system  of 
units  employed,  we  may  write 

W/M  =  g;     or     W  =  Mg. 

126.  We  will  consider  the  experimental  evidence  for  the 
Second  Law  after  we  have  discussed  the  Third  Law,  meanwhile 
observing  that  both  of  them  are  abundantly  verified  by  the 
calculations  found  in  the  Nautical  Almanac.  But  before  leaving 
the  Second  Law,  we  must  note  two  important  facts  of  experience 
not  explicitly  stated  by  it,  but  implied  by  its  form. 

(1)  It  says  nothing  about  the  existing  state  of  motion  of  the 
body  acted  on.  The  effect  of  a  force  in  producing  new  velocity 
is  found  to  be  the  same  whether  the  body  is  at  rest  or  already 
moving  at  high  speed. 

Set  two  small  objects  on  the  edge  of  a  table  arid  sweep  a 
heavy  paper-knife  along  the  table  so  as  to  strike  one  of  them 
horizontally  towards  the  end  of  the  room,  while  the  other  is 
merely  dislodged.  They  will  be  heard  to  strike  the  floor  at  the 
same  moment,  the  one  at  the  foot  of  the  table,  the  other  many 


128  MECHANICS  [CHAP. 

feet  away.  If  a  rifle  were  fired  horizontally  so  as  just  to  dislodge 
an  object  at  its  muzzle,  the  bullet  and  the  object  would  still 
strike  the  earth  at  the  same  moment,  though  perhaps  many 
hundred  feet  apart.  Mere  speed  in  no  way  exempts  a  body  from 
the  action  of  gravity  or  any  other  force. 

(2)  The  Second  Law  makes  no  mention  of  any  other  forces 
that  may  be  acting  on  the  body  at  the  same  time.  It  is  found 
that  every  force,  however  small,  produces  its  whole  effect,  however 
great  may  be  the  other  forces  acting  on  the  body.  The  attraction 
of  a  falling  stone  upon  the  earth  has  its  full  effect  in  modifying 
the  earth's  motion,  although  the  earth  is  subject  at  the  same 
time  to  the  enormous  attraction  of  the  sun.  Its  motion  is  com- 
pounded of  all  those  produced  by  the  forces  acting  on  it,  including 
the  slight  pull  of  the  stone. 

Newton  was  the  first  to  state  this  explicitly  in  a  corollary  to 
the  Second  Law  In  his  tract  Propositions  de  Motu,  which 
preceded  the  Principia,  he  says: 

"Corpus  in  dato  tempore  viribus  conjunctis  eo  ferri  quo 
viribus  diversis  in  temporibus  aequalibus  successive." 

If  two  forces  act  simultaneously  on  a  body,  and  if  they  would 
respectively  produce  the  mo- 
tions AB,  AC,  when  acting 
separately  for  the  same  interval 
of  time,  then  the  body  will 
move  in  that  interval  to  D, 
since  the  forces  and  the  motions 
produced  by  them  are  indepen-  D 

dent  of  each  other. 

In  the  Principia  the  first  Corollary  to  the  Laws  of  Motion 
stands  thus :  "  Corpus  viribus  conjunctis  diagonalem  parallelo- 
grammi  eodem  tempore  describere,  quo  latera  separatist 

The  forces  are  supposed  to  be  single  impulses  applied  at  A,  in 
the  directions  AB,  AC  respectively,  and  sufficient  to  carry  the 
body  to  B  and  C  in  equal  times.  The  body  will  arrive  at  D,  and 
must  have  come  by  the  straight  line  AD,  by  Law  I,  for  once 
started,  it  is  not  acted  on  by  any  force. 

In  Corollary  2,  Newton  points  out  that  Statics  may  be  deduced 


XIII]  THE   THIRD   LAW    OF    MOTION  129 

from  this  principle,  and  illustrates  it  by  deducing  the  Principle  of 
the  Lever. 

This  is  the  first  distinct  formulation  of  the  Parallelogram  of 
Forces.  We  shall  return  to  its  formal  proof  later. 

127.  By  the  Second  Law  we  can  calculate  the  motion  of  any 
body  when  we  know  its  mass  and  the  forces  acting  on  it.     But 
these  forces  are  applied  from  without  by  other  bodies.     They  are 
pushes,  or  pulls,  or  attractions,  and  there  is  always  a  reaction 
upon  the  body  that  pushes  or  pulls.    What  is  the  relation  between 
the  forces  mutually  exerted  upon  each  other  by  any  two  bodies  ? 
Newton's  answer  to  this  question  is  given  in  his  Third  Law. 

Law  III.  To  every  Action  there  is  an  equal  and  opposite 
Reaction. 

Pressure  and  counter-pressure,  action  and  counter-action  are 
equal.  All  force  is  of  the  nature  of  a  stress,  that  is,  a  mutual 
action  between  two  bodies,  the  same  from  whichever  side  it  is 
looked  at. 

You  cannot  exert  a  pressure  unless  you  meet  with  a  resistance, 
and  then  the  pressure  and  the  resistance  grow  side  by  side,  being 
always  equal.  You  cannot  cut  a  piece  of  paper  with  one  blade  of 
a  pair  of  scissors,  nor  crack  a  nut  with  one  half  of  a  pair  of 
nutcrackers.  You  cannot  drive  a  nail  into  a  board  unless  it  is 
supported  behind,  for  the  board  yields  before  the  pressure  is  great 
enough  to  force  the  nail  in.  A  cannon  ball  can  exert  no  force  till 
it  meets  with  an  obstacle,  and  then  only  so  great  a  force  as  that 
with  which  it  is  resisted.  You  cannot  pull  an  object  harder  than 
it  pulls  back. 

128.  Beginners  are  liable  to  a  difficulty  in  admitting  this. 
They  say :  "  Why  then  does  any  object  ever  succeed  in  moving 
any  other  at  all  ? "     Newton  himself  considers  the  case  of  a  horse 
drawing  a  cart.     No  more  instructive  problem  can  be  found,  so  we 
will  examine  it  in  some  detail. 

According  to  the  Third  Law  the  cart  pulls  the  horse  backwards 

as   hard  as   the    horse   pulls   the   cart    forwards.     Everyone   will 

admit  that  if  a  spring  dynamometer  is  employed  to  measure  the 

tension  of  the  traces,  it  does  not  matter  which  end  is  attached  to 

c.  9 


130 


MECHANICS 


[CHAP. 


the  horse,  and  which  to  the  cart.  The  reading  will  be  the  same. 
Then  how  is  it  that  they  ever  get  into  motion  ? 

The  difficulty  brings  out  a  very  important  point.  In  attacking 
any  mechanical  problem  it  is  essential  to  begin  by  fixing  upon  the 
system  whose  rest  or  motion  we  are  going  to  consider.  We  may 
make  it  include  one  body,  or  a  collection  of  bodies,  or  the  whole 
universe;  but  we  must  be  clear  as  to  what  it  is. 

In  the  case  of  the  horse  and  cart ;  is  it  wished  to  know  why 
the  cart  advances  ?  or  why  the  horse  advances  ?  or  why  they  both 
advance  ? 

Let  us  begin  with  the  cart.  The  cart  is,  then,  the  system 
whose  motion  is  to  be  determined.  It  is  to  be  supposed  isolated 
in  our  minds  from  all  other  objects,  as  in  an  imaginary  enclosure 
acbd. 


Fig.  60. 

Let  M  be  its  mass.  Let  us  now  go  round  the  enclosure  and 
see  what  forces  act  on  it.  When  we  know  the  forces  and  the 
mass,  the  Second  Law  of  Motion  will  tell  us  what  will  be  its 
movements. 

(1)  The  earth  exerts  two  forces  on  the  cart,  (a)  an  attraction, 
which  is  the  weight  W  of  the  cart ;  and  (b)  an  upward  pressure  P  at 
the  point  where  the  wheel  rests  on  the  ground.  To  each  of  these, 
by  Law  III,  there  is  an  equal  and  opposite  force  exerted  by  the 


Xlll]          PROBLEM  OF  THE  HORSE  AND  CART  131 

cart  upon  the  earth.  But  with  these  latter  forces  we  are  not  con- 
cerned at  the  moment,  since  we  are  not  considering  the  motion  of 
the  earth,  but  of  the  cart. 

The  upward  pressure  (b)  is  also  equal  to  the  downward  pull  of 
the  weight  (a) ;  for  if  it  were  greater  than  the  weight,  the  cart 
would  rise  into  the  air,  and  if  it  were  less,  the  cart  would  sink 
into  the  ground,  as  in  fact  it  does  on  soft  ground,  till  the  resistance 
becomes  equal  to  the  weight.  These  forces  therefore  balance,  and 
may  be  left  out  of  our  account. 

(2)  The  tension  of  the  traces  acts  forwards.     (Observe  that 
there  is  never  any  difficulty  in  deciding  which  way  a  tension  or 
pressure  acts,  if  we  are  clear  as  to  which  is  the  body  whose  motions 
we  are  studying.) 

Let  the  tension  be  T. 

(3)  The  only  other  forces   acting   on   the   cart   are  certain 
frictions  and  resistances,  at  the  axles,  on  the  ground,  and  against 
the  air.     For  simplicity,  let  us  suppose  that  these  are  equivalent 
to  a  direct  pull  by  a  rope,  backwards,  of  magnitude  R. 

Now  provided  the  tension  T  be  greater  than  the  resistance  R, 
there  will  be  a  balance  of  force  T  —  R  forwards,  and  the  cart  will 
begin  to  move,  for  it  will  be  subject  to  an  acceleration  forwards, 

*-T-; W 

Next  consider  the  horse.  Let  his  mass  be  m,  and  draw  an 
imaginary  enclosure  round  him  as  before.  The  forces  acting  on 
him  through  this  enclosure  are : 

(1)-  His  weight,  and  the  upward  pressure  of  the  earth,  which 
so  long  as  he  stands  still  will  balance  as  in  the  case  of  the  cart. 

(2)  If  he  tries  to  go  forwards,  there  will  be  a  tension  of  the 
traces  pulling  him  backwards ;  and  this,  by  the  Third  Law,  will 
be  exactly  equal  to  T,  the  forward  tension  of  the  traces  upon  the 
cart. 

But  where  is  the  external  force  which  is  required  by  Law  II 
to  make  him  go  forwards?  He  cannot  exert  it  upon  himself. 
This  is  implied  by  Law  II,  and  explicitly  stated  by  Law  I.  No 
man  can  raise  himself  by  pulling  at  his  own  boot-straps,  or  if 

•  9—2 


132  MECHANICS  [CHAP. 

seated  on  a  chair,  by  lifting  at  the  lower  rails.  It  must  come 
from  outside,  from  some  other  object. 

Accordingly  the  horse,  besides  maintaining  the  downward 
pressure  necessary  to  support  his  weight,  thrusts  the  earth  back- 
wards; and  by  Law  III  the  earth  immediately  thrusts  him 
forwards  with  an  equal  reaction.  Let  this  be  F.  So,  too,  the 
skater,  gripping  the  ice  with  the  sharp  edge  of  his  skate,  gives  a 
back-thrust,  and  is  himself  driven  forwards;  and  the  swimmer 
advances  by  thrusting  back  a  wedge  of  water  from  between  his 
legs. 

If  the  horse  thrusts  back  hard  enough,  F  will  be  greater  than 
T ;  and  he  will  begin  to  move  with  an  acceleration 

a  -*=*  (2) 

m    

Thus  although  the  back -pull  on  the  horse  is  the  same  as  the 
forward  pull  on  the  cart,  both  will  advance  provided  that  F  is 
greater  than  T,  and  T  than  R. 

Assuming  that  we  know  the  force  with  which  the  horse 
thrusts,  F',  the  resistances  to  motion,  R ;  and  the  masses  My  m\ 
we  ought  to  be  able  to  solve  the  problem  completely,  and  find  at 
what  rate  motion  will  ensue.  But  we  have  at  present  only  two 
equations  to  find  three  unknowns,  viz.  the  two  accelerations,  and 
the  tension  of  the  traces.  It  will  be  found  that  there  is  always  a 
dynamical  equation  for  each  of  the  accelerations.  If  no  unknown 
reaction,  such  as  the  tension,  comes  in,  this  will  be  sufficient. 
But  wherever  a  reaction  of  this  kind  occurs,  it  must  be  in  con- 
sequence of  some  connection  between  the  parts  of  the  system, 
whereby  they  have  to  move  in  a  certain  definite  relation  to  each 
other.  That  is,  for  every  unknown  reaction  there  will  be  a  geo- 
metrical equation,  expressing  the  special  relation  of  the  parts  of 
the  system  which  gives  rise  to  the  reaction.  Thus  there  will 
always  be  enough  equations  to  ensure  a  definite  solution. 

In  the  present  case,  so  long  as  the  traces  hold,  the  horse  and 
cart  have  to  advance  together,  so  that  the  two  accelerations  are 
equal.  Hence 

a1  =  a2 (3). 

F-T T-R 


Xlll]  SOLUTION   OF   A   DYNAMICAL   PROBLEM  133 

m     MF+mR 

Whence  T  =     ,,         - , 

M  +  m 

and  by  substitution, 

MF+mR 
T-R        M+m 


M  M 

=  F-R 

M  +  m' 

Substitution  in  (2)  would  of  course  give  the  same  value  for  aa. 

Finally,  let  us  consider  the  horse  and  cart  as  one  system.  The 
imaginary  enclosure  must  now  be  drawn  round  the  two  together. 
The  only  unbalanced  forces  acting  on  the  system  from  without 
will  now  be  the  forward  thrust  of  the  earth,  F,  and  the  back-pull 
of  the  resistances,  R.  The  weights  and  the  upward  thrusts  of  the 
earth  balance  as  before. 

The  mass  to  be  acted  upon  is  the  total  mass  of  the  horse  and 
cart.  Hence  the  acceleration  of  the  whole  system  will  be 

F-R 


The  former  result  can  thus  be  written  down  directly  when  a 
knowledge  of  the  tension  is  not  required. 

What  has  become  of  the  tension  in  this  case  ?  It  is  no  longer 
an  external  force,  but  a  mere  internal  reaction  between  two  parts 
of  the  system,  which  can  no  more  affect  its  motion  as  a  whole  than 
any  of  the  other  forces  which  hold  the  parts  of  the  cart  and  .the 
frame  of  the  horse  together.  It  does  not  therefore  enter  into  the 
equations.  On  the  other  hand,  the  tension  can  only  be  found  by 
considering  the  motion  of  the  two  parts,  between  which  it  occurs, 
separately. 

129.  The  solution  of  any  dynamical  problem  by  Newton's 
method  involves  no  principle  which  has  not  been  illustrated  in 
the  elementary  question  we  have  been  discussing.  The  rest  is 
mere  elaboration  of  mathematics  to  meet  the  complexities  in- 
troduced by  changes  in  the  direction  of  motion,  or  of  force  in 
curved  orbits,  or  the  number  of  bodies  or  parts  interacting  on  each 
other.  The  process  consists  of: 


134  MECHANICS  [CHAP. 

(1)  writing   down   the   Equations   of  Motion,  i.e.  equations 
expressing  the  accelerations  of  the  different  bodies  concerned  in 
terms  of  their  masses  and  the  forces  acting  on  them  ; 

(2)  writing    down   the   geometrical    equations   defining   the 
relative  movements  of  parts  which  exert  reactions  on  each  other ; 
and 

(3)  solving  the  equations. 


EXAMPLES. 

1.  Shew  that  if  a  train  travelling  at  60  miles  an  hour  is  suddenly  brought 
to  'rest  by  a  head-on  collision,  a  passenger  facing  the  engine  will  strike  the 
opposite  wall  of  the  carriage  as  if  he  had  fallen  on  to  it  from  the  top  of  a  tower 
121  feet  high. 

2.  A  rain  drop  experiences  a  resistance  from  the  air  proportional  to  the 
square  of  its  velocity.     Hence  shew  that  there  is  a  limit  to  the  velocity  it  can 
acquire,  and  that  after  this  terminal  velocity  is  reached  the  drop  falls  with 
uniform  velocity. 

3.  Find  the  acceleration : 

(1)  in  foot-second  units  when 

(a)  a  force  of  12  poundals  acts  on  a  mass  of  2  Ibs, 

(b)  a  force  of  14  pounds  weight  acts  on  112  Ibs. 

(2)  in  c.G.s.  units  when 

(a)  a  force  of  10,000  dynes  acts  on  a  mass  of  80  gms. 
(6)    a  force  of  10  pounds  weight  acts  on  20  kilograms. 

4.  What  force  (in  poundals)  is  required  to  produce  an  acceleration  of 
12  ft. -sec.  units  in  masses  of  2  Ibs. ;  12  ozs. ;  112  Ibs.?    Express  these  forces  in 
pound-weights. 

5.  How  many  dynes  must  act  on  2  kilograms  to  produce  an  acceleration 
of  8000  C.G.s.  units?    Express  this  force  in  gramme-weights. 

6.  Find  the  mass  acted  on  when  : 

(1)  a  force  of  3  pounds  weight  causes  an  acceleration  of  1,  36,  and  96 
British  units  respectively ; 

(2)  a  force  equal  to  the  weight  of  10  gms.  causes  an  acceleration  of  196 
C.G.s.  units. 

7.  A  force  equal  to  the  weight  of  3  Ibs.  acts  on  a  mass  of  2  kilograms  for 
10  seconds.     Find  the  velocity  acquired  and  the  distance  travelled,  in  C.G.S. 
units. 


XIII]  KINETICS  135 

8.  An  engine  pulls  a  train  of  200  tons  mass  with  a  force  equal  to  the 
weight  of  a  ton  and  a  half.     Find  the  acceleration ;  the  speed  acquired  (a)  in 
half  a  minute,  and  (6)  in  half  a  mile ;  the  distance  run  in  5  minutes  j  and  the 
time  required  to  run  the  first  two  miles. 

9.  A  train  of  120  tons  mass  running  45  miles  an  hour  is  pulled  up  by  the 
brakes  in  half  a  mile.     What  is  the  retarding  force  of  the  brakes  in  ton- 
weights  ? 

10.  A  train  has  its  speed  reduced  from  40  to  20  miles  an  hour  in  400 
yards.     What  is  the  resistance  of  the  brakes  in  pounds  per  ton-weight  of  the 
train  1    How  far  and  how  long  will  the  train  continue  to  run  before  it  stops  ? 

11.  What  force  is  required  to  stop  a  train  of  100  tons  going  30  miles  an 
hour  (1)  in  half  a  minute,  (2)  in  half  a  mile  1 

12.  A  one-ounce  bullet  is  fired  from  a  rifle  barrel  4  feet  long  with  a  speed 
of  1600  feet  per  second.     What  was  the  average  pressure  of  the  powder  on  the 
bullet? 

13.  A  600  Ib.  cannon  ball  is  fired  from  a  gun  weighing  12  tons  with  a 
velocity  of  2000  feet  per  second.     If  the  gun  is  free  to  move,  with  what 
velocity  will  it  start  backwards? 

(By  the  Third  Law  the  pressure  on  the  gun  must  throughout  the  explosion 
be  equal  and  opposite  to  that  on  the  shot.  Hence  the  momentum  of  the  gun 
must  be  equal  and  opposite  to  that  of  the  shot.) 

14.  A  half-ounce  bullet    in   passing  through    a  2-inch  plank  has  its 
velocity  reduced  from  900  to  600  feet  per  second.     Find  its  acceleration  in  the 
plank,  and  hence  the  average  resistance  offered  to  it.     What  thickness  of 
plank  would  have  just  stopped  it  ? 

15.  A  curling  stone  is  projected  along  ice  with  a  velocity  of  48  feet  per 
second.     Express  the  resistance  due  to  friction  as  a  decimal  of  the  weight  of 
the  stone,  if  it  stops  (1)  in  300  feet,  (2)  in  15  seconds. 

16.  A  slinger  using  a  sling  2  ft.   6  in.  long  whirls  a  stone  weighing 
2  ounces  in  a  nearly  horizontal  circle  above  his  head,  so  as  to  make  5  revolu- 
tions per  second.     What  is  the  tension  of  the  string? 

(Calculate  first  the  acceleration  of  the  stone  to  the  centre  by  §  77.) 

17.  How  many  times  faster  must  the  earth  turn  on  its  axis  so  that 
objects  at  the  equator  should  just  lose  their  weight? 

18.  If  a  train  of  200  tons  were  to  run  at  60  miles  an  hour  at  the  equator, 
what  difference  would  it  make  in  the  pressure  on  the  rails  if  the  train  were  to 
run  due  west  instead  of  due  east  ? 

19.  A  bicyclist  is  running  at  20  miles  an  hour  round  a  circular  track  with 
four  laps  to  the  mile.     What  should  be  the  inclination  of  his  machine  to  the 
vertical  ? 

20.  A  fly-wheel  with  a  rim  weighing  10  tons  has  a  diameter  of  6  feet,  and 
runs  at  160  revolutions  per  minute.     If  there  are  six  spokes,  what  pull  has 
each  spoke  to  bear  in  preventing  the  rim  from  flying  to  pieces? 


CHAPTER    XIV. 

EXPERIMENTAL  VERIFICATION  OF  THE   LAWS  OF  MOTION. 
ATWOOD'S  MACHINE. 

130.     THE  Laws  of  Motion  may  be  tested  directly  by  means  of 

Atwood's         Atwood's  Machine.    This  consists  of  a  light  pulley 

Machine.         Qf  aiummium^  balanced  very  truly  on  its  axis,  and 

supported  on  a  set  of  friction  wheels  to  reduce  the  friction  as  much 

as  possible.     A  cord  passes  over  it  and  carries  two  equal  weights, 

one  of  which  runs  in  front  of  a  vertical  scale.     At  the  zero  of  the 

scale  is  a  hinged  platform  which  is  pulled  flat  against  the  scale 

by  a  spring  on  withdrawing  a  catch.     A  clock  beating  seconds  is 

arranged  to  withdraw  the  catch  electrically  at  the  instant  when 

the  minute  hand  passes  the  mark  60"  at  the  top  of  the  dial. 

On  the  scale  are  two  moveable  platforms,  the  upper  one  being 
a  hollow  ring  through  which  the  weight  can  pass,  the  lower  a  solid 
plate  which  brings  it  to  rest.  Small  weights  are  provided,  con- 
sisting of  flat  brass  slips  which  can  rest  on  the  top  of  the  main 
weights,  but  are  too  long  to  pass  through  the  ring-platform. 

One  of  the  Atwood  Machines,  figure  61,  is  for  use  in  Lecture 
Demonstrations.  It  has  a  scale  of  750  centimetres  ;  and  to  equalize 
the  weight  of  thread  on  each  side,  an  idle  thread  hangs  in  a  loop 
from  the  bottoms  of  the  moving  weights.  This  also  gives  the 
experimenter  control  of  the  machine  from  below. 

The  other  Machine,  figure  61  a,  constructed  by  the  Cambridge 
Scientific  Instrument  Company,  is  for  accurate  work.  It  has  a  steel 
scale/with  geometrical  contact  adjustments  for  the  platforms  and  fine 


CHAP.  XIV] 


ATWOOD'S  MACHINE 


137 


screw  motions  for  setting 
them  accurately.  The  times 
of  release  and  of  passing 
each  of  the  platforms  are 
recorded  electrically  on  a 
chronograph,  and  the  mov- 
ing weight  is  brought  to 
rest  in  a  dashpot  at  the 
bottom  of  the  scale  column. 
A  weight  suspended  by  a 
spiral  spring  vibrates  verti- 
cally, giving  seconds  on  the 
chronograph. 

131.  When    only    the 
First  Law      *wo  equal   large 

of  Motion.        weights  are  uge(J) 

there  is  no  unbalanced  force 
tending  to  set  up  motion. 
They  will  be  found  to  rest 
in  any  position,  and,  if  set 
in  motion,  they  travel  uni- 
formly, passing  over  the 
same  number  of  centimetres 
in  every  second,  except  for 
an  almost  inappreciable  loss 
of  speed  through  friction. 

132.  Place  one  of  the 
second  Law   small  flat  weigh ts 
of  Motion,     across  the  top  of 
the  large  weight.     Let  the 
masses  of  the  large  and  small 
weights  be  M  and  m.     The 
only  unbalanced  force  acting 
on  the  system  is  the  weight 
of  m. 

The  total  mass  to  be  set 
in  motion  is  2M  +  m. 


Fig.  61. 


138 


MECHANICS 


[CHAP. 


Fig.  61  a. 


XI  V]  ATWOOD'S    MACHINE  139 

By  the  Second  Law  of  Motion  there  should  be  an  acceleration 

As  this  is  a  constant  acceleration,  the  distance  run  in  t  seconds 

at2 
from  rest  will  be  s  =  -=-  ,  and  the  velocity  acquired  v  =  at. 

If  the  weights  have  been  weighed  beforehand,  a  may  be 
calculated,  and  the  values  of  s  and  v  found  for,  say,  3  seconds.  The 
ring-platform  should  be  set  at  the  distance  s  below  the  zero,  and 
the  lower  platform  near  the  bottom.  Pass  the  weight  carrying  the 
small  weight  above  the  top  platform  ;  set  the  platform  and  lower 
the  weight  on  to  it.  On  starting  the  clock  the  platform  will  be 
heard  to  fall  at  the  beginning  of  a  minute,  and  the  cross  weight 
will  be  caught  by  the  ring  almost  exactly  at  the  third  following 
tick.  The  seconds  should  then  be  counted  till  the  last  platform  is 
reached. 

After  passing  the  ring-platform  the  system  is  not  acted  on  by 
any  unbalanced  force,  so  that  its  velocity  is  uniform.  It  may  be 
found  by  dividing  the  distance  between  the  two  lower  platforms 
by  the  number  of  seconds  required  to  pass  from  the  one  to  the 
other.  By  repeated  trials  the  platforms  may  be  set  so  that  the 
passage  of  the  weights  through  them  coincides  with  a  tick  of  the 
clock,  and  no  fractions  of  a  second  need  be  estimated. 

Besides  moving  the  large  weights  and  itself,  the  small  weight 
has  to  set  in  motion  the  aluminium  pulley.  This  can  be  allowed 
for  by  supposing  that  an  extra  small  mass  x  must  be  set  moving, 
so  that 


Using    a    different    small   weight   ra',   we    find    a    different 
acceleration, 

,  m'g  . 


"ZM+m' 

a  and  of  may  be  determined  by  experiment  with  the  machine,  and 
x  is  then  found  from  (1)  and  (2). 

A  third  experiment  should  be  tried  with  still  another  weight 
ra".  The  formula  with  the  value  of  x  substituted  in  it  should  then 
give  correct  results  for  s  and  v. 


140  MECHANICS  [CHAP. 

133.     The  weights  and  the  middle  platform  are  removed  from 
the  machine.     The  top  platform  is  set,  and  any 

To  find  the  .  r  .  * 

Dynamical  unit  of  small  object  placed  upon  it.  The  lower  platform 
is  placed  at  490  cm.,  and  the  clock  is  started.  The 
small  object  is  heard  to  strike  the  platform  exactly  at  the  first  tick 
after  the  top  platform  falls.  Its  average  velocity  must  be  490  cm. 
per  second,  and  therefore  the  velocity  at  the  end  of  one  second  is 
980  cm.,  or  82  -2  feet  per  second. 

Thus  Galileo's  experiments  at  the  tower  of  Pisa  may  be  verified  ; 
and  since  the  object  might  have  been  a  gramme  or  pound  weight, 
the  unit  force  must  be  the  1 /980th  part  of  a  gramme  weight,  or 
the  l/32'2th  part  of  a  pound  weight  according  to  the  system  of 
units  adopted.  (§  124.) 


134.     The  problem  of  Atwood's  machine  is  the  same  as  that  of 

Third  Law  of      ^e  horse  and  cart  (§  128),  with  the  descending 

Motion.  weights  substituted  for  the  horse,  the  ascending 

weight  for  the  cart,  and  the  string  for  the   traces.     We   have 

treated  the  weights  as  one  system,  since  only  the  acceleration  was 

needed.     If  the   tension    T  of  the   string  is   to   be   found,  the 

accelerations  a  and  of  must  be  calculated  separately.     Thus 

.......  (1)' 


M  +  m 
T-Mg 


a  =  a?  .......................................  (3). 

Whence  T_*(M+*)lfg 

2M  +m 

•t  tna 

a=mr^> 

as  before. 

In  arriving  at  this  correct  result,  we  have  assumed  that  the 
string  transmits  equal  and  opposite  action  and  reaction  T  between 
the  ascending  and  descending  weights,  so  that  the  Third  Law  is 
verified. 

The  advantage  of  Atwood's  Machine  is  that  it  enables  us  to 


xiv]  ATWOOD'S  MACHINE  141 

work  with  a  constant  acceleration  which  may  be  any  small  fraction 
of  gravity  we  choose  according  to  the  value  of  m/(2M  +  m).  It  is 
easier  to  observe  the  slower  motion,  and  the  two  platforms  afford 
convenient  means  of  measuring  the  acceleration  during  the  first 
part  of  the  motion,  as  well  as  the  final  velocity  acquired. 


EXAMPLES. 

1.  The  large  weights  in  an  Atwood's  Machine  are  each  223  gms.     The 
rider  is  13 '4  gms.     How  many  centimetres  below  the  stage  must  the  ring- 
platform  be  set  so  as  to  be  reached  in  2  seconds  ?     How  far  below  this  must 
the  last  platform  be  set  so  as  to  be  reached  8  seconds  afterwards  ? 

2.  With  the  above  weights  the  ring-platform  is  set  so  as  to  be  reached  in 
3  seconds,  when  it  is  found  that  the  lowest  platform  must  be  set  429  cms. 
below  it,  in  order  to  be  reached  5  seconds  afterwards.    Hence  find  the  value 
of  gravity. 

3.  In  an  experiment  with  the  weights  in  Question  1,  it  is  found  that  the 
ring-platform  must  actually  be  set  at  55-3  cms.  below  the  zero  in  order  to  be 
reached  in  2  seconds.     On  the  assumption  that  the  effect  of  the  mass  of  the 
pulley,  which  has  also  to  be  set  in  motion,  is  equivalent  to  the  permanent 
addition  of  a  small  mass  x  to  the  two  large  weights,  an  experiment  was  tried 
with  another  rider  weighing  27  gms.     The  platform  had  then  to  be  set  at 
108-3  cms.  to  be  reached  in  2  seconds.     From  these  two  experiments  find  x 
and  the  value  of  gravity. 

4.  A  mass  of  10  Ibs.  is  placed  on  a  smooth  table  and  connected,  by  a 
string  passing  over  a  smooth  pulley  at  the  edge,  with  a  weight  of  2  Ibs. 
hanging  freely.     Find  the  acceleration  of  the  system.     How  far  from  the  edge 
must  the  10  Ib.  weight  be  placed  so  as  to  fall  off  in  2  seconds  ? 

5.  Find  the  tension  of  the  string  in  Question  1  (before  and  after  the  ring- 
platform  is  reached)  and  in  Question  4. 


CHAPTER  XV. 

WORK  AND  ENERGY. 

135.  THE  Newtonian  method  of  solving  dynamical  problems 
begins  by  fixing  the  rate  at  which  the  velocity  changes,  i.e.  the 
acceleration,  by  means  of  the  formula  a  =  P/M. 

From  this  it  proceeds  to  calculate  what  will  be  the  velocity, 
and  where  the  body  will  be  at  every  succeeding  instant,  by  the 
kinematical  formulae : 

v  =  at  and  s  =  atf/2, 
or  their  proper  modifications  if  the  acceleration  is  variable. 

136.  For  many  purposes  we  do  not  need  to  follow  the  motion 
with  such  intimate  scrutiny,  but  are  content  to  fix  our  attention 
on  the  final  velocity  acquired  when  some  other  position  is  reached, 
disregarding  what  has  gone  on  meanwhile.     This  we  can  do  by 
means  of  the  third  kinematic  formula 

v*/2  =  as, 

or  its  dynamical  equivalent  obtained  by  multiplying  both  sides  by 
the  mass  M.     Thus 

Mv*l2  =  Mas  =  Pxs  (1). 

This  formula  tells  us  what  speed  will  be  acquired  by  a  mass 
M,  when  a  force  P  pushes  it  steadily  through  a  distance  s ;  or,  if 
the  mass  be  already  in  motion  with  given  velocity  v,  and  the  force 
P  be  employed  to  stop  it,  it  tells  us  how  far  the  force  will  be 
driven  back,  before  the  body  is  brought  to  rest,  viz. 

s  =  Mv*/2P. 

The  product  P  x  s  on  the  right-hand  side  of  (1)  has  already 
received  a  name.  It  is  the  work  done  by  the  force  P  while  the 
body  travels  the  distance  s.  (§  45.) 


CHAR  XV]  WORK   AND   ENERGY  143 

137.  A  body  or  system  on  which  work  has  been  done  is  found 
to   have   an    increased   power   of  doing   work   itself,  that   is,  of 
producing  physical  changes  in  other  bodies.     It  is  therefore  said 
to  possess  more  energy  than  before.     This  energy  may  take  two 
forms. 

138.  The  body  may  have  been  set  in  motion  with  regard  to 
i.   Energy  of     other  bodies.     By  the  first  law  of  motion  it  will 

then  continue  in  motion  until  some  resisting  force 
is  employed  to  stop  it.  Let  it  be  brought  to  rest  by  the  constant 
force  P  in  a  distance  s.  Before  it  stops,  it  does  a  quantity  of  work 
P  x  s,  and  since 


it  appears  that  the  work  done  by  a  body  of  mass  M  and  velocity  v 
before  it  can  be  brought  to  rest  is  measured  by  Mv2/2. 

The  quantity  of  work  which  the  moving  body  can  do  in  virtue 
of  its  motion  is  called  its  Kinetic  Energy,  or  energy  of  motion,  and 
is  measured  by 


139.     That  motion  confers  on  a  body  a  certain  "efficacy"  or 

power  of  overcoming  opposing  forces,  and  producing 

Energy  and        changes  in  other  bodies  was  of  course  a  familiar 

fact.    For  a  long  time  a  controversy  raged  between 

the  followers   of  Des  Cartes   and  Leibnitz   as   to  whether  this 

"  efficacy  "  of  the  body  was  proportional  to  the  velocity  or  to  the 

square  of  the  velocity.     The  dispute  is  seen  to  be  meaningless  if 

we  compare  the  Newtonian  formula, 

MV  Pt, 

(Momentum)         (Impulse) 
with  the  energy  formula,  first  employed  by  Huyghens, 


(Energy)  (Work  done) 

Given  a  mass  M  moving  with  a  velocity  V.  The  first  formula 
tells  us  how  long  (t  =  MV/P),  the  second  how  far  (s  =  MV2/2P)  it 
will  continue  to  move  against  a  given  force  P,  before  it  can  be 
brought  to  rest. 


144  MECHANICS  [CHAP. 

Conversely  the  two  formulae  respectively  give  the  time  and 
the  distance  required  by  the  force  P  to  produce  in  M,  initially  at 
rest,  the  velocity  F. 

Thus  both  the  Cartesians  and  the  Leibnitzians  were  right. 
The  time  during  which  a  moving  body  will  go  on  overcoming 
a  given  resistance  is  proportional  to  its  velocity,  the  distance  to 
the  square  of  its  velocity.  A  body  thrown  vertically  upwards,  and 
then  again  with  twice  the  speed,  will  in  the  second  case  rise  for 
twice  as  long  a  time,  but  four  times  as  high  as  before. 

A  moving  body  is  just  a  moving  body,  and  its  effects  can  be 
calculated  when  we  know  its  mass  and  its  velocity.  In  making 
the  calculations  we  often  have  to  reckon  the  product  Mv\  often 
again  the  product  Mv*/2  ;  so  often,  in  fact,  that  it  is  worth  while 
to  denote  them  by  the  special  names  Momentum,  and  Energy. 
But  it  must  not  be  supposed  that  these  are  occult  properties  of 
the  body,  which  may  sometimes  exhibit  one,  and  sometimes  the 
other.  Every  moving  body  has  both  Momentum  and  Energy,  i.e. 
we  can  calculate  both.  MV  and  MV2/2  for  it.  Which  of  the  two 
we  should  choose,  depends  on  the  purpose  we  have  in  view. 

140.  Let  us  compare  in  respect  of  momentum  and  energy  an 
ironclad  of  ten  thousand  tons  mass,  so  nearly  at  rest  that  it  is 
moving  only  one  inch  per  second,  with  a  one-ounce  bullet  moving 
1600  feet  per  second. 

(1)  Momentum  -  MV.    This  is 

For  the  ironclad  2240  x  10,000  x  1/12  =  1,866,666-6  British  units 
„       bullet       1/16x1600  =100 

(2)  Energy  =  MF2/2. 

1  ft  ftftft  v  994ft 

For  the  ironclad  -      ~~     -  x  (1/12)2  =  77,777"?  foot-poundals. 


„      bullet  x  =80,000       „ 

The  ironclad  has  enormously  greater  momentum  than  the 
bullet  ;  but  the  bullet  has  rather  the  greater  energy. 

To  understand  precisely  what  this  means,  let  us  suppose  that 
each  has  to  be  brought  to  rest  by  holding  against  it  a  perfectly 
hard  shield  with  a  steady  force  of,  say,  1000  poundals.  We  may 


XV]  WORK   AND   ENERGY  145 

ask;     (1)    How  long  will  each  go   on  pushing  the  shield  back? 
Or    (2)    How  far? 

/n\     a-         j/rTT-      Txt    xi.      f  momentum 

(1)     Since  MV=  Pt;  therefore  t  = 

force 

For  the  ironclad  t  =  1>8616'^6'6     =  1,866'6  seconds. 
1UUU 


„      „    bullet     *  =  j5  =1/10  second. 

(2)    Since  MV2/2  =  P.s',  therefore  s  = 


force 


77  777'? 

For  the  ironclad  s  =      '  =  77'1?  feet. 

1UUU 


„ 


bullet     ,  = 


The  bullet  will  be  brought  to  rest  in  the  tenth  part  of 
a  second,  whereas  the  ironclad  will  creep  on  for  more  than  half 
an  hour.  Nevertheless  the  bullet  will  push  the  shield  back 
through  80  feet,  as  against  77*7  feet  for  the  ironclad. 

This  is  easily  understood.  For  the  average  speed  of  the 
bullet,  which  decreases  uniformly  from  1600  feet  per  second  to 
nothing,  is  800  feet  per  second.  That  of  the  ironclad  is  half  an 
inch  per  second.  The  bullet  in  one-tenth  of  a  second  travels  its 
80  feet,  while  the  ironclad  only  creeps  over  77'1?  feet  in  its  half 
hour. 

141.     Although  a  body  on  which  work  has  been  done  may  be 

ii    Energy  of    at  rest>  there   mav   have   been   a   change  in  its 

Position.  position  with  regard  to  other  bodies ;  or  in  its 

shape,  that  is,  the  position  of  its  own  parts  with  regard  to  each 

other.     And  in  consequence  of  this  change  it  may  be  capable  of 

doing  more  work  than  before,  i.e.  possess  more  energy. 

To  raise  a  one-pound  clock-weight  through  a  height  of  three 
feet,  three  foot-pounds  of  work  must  be  done.  The  weight  has 
then  been  drawn  apart  a  distance  of  three  feet  from  the  earth 
against  the  attraction  of  gravity.  The  "  system,"  consisting  of  the 
earth  and  the  weight  separated  from  it,  has  the  power  of  doing 
this  work  back  again ;  for,  if  allowed  to  run  down,  the  weight  will 
do  so,  and  drive  the  clock  for  a  long  time.  The  three  foot-pounds 
c.  10 


146  MECHANICS  [CHAP. 

of  work  have  thus  been  stored  up  in  the  system,  which  possesses 
three  foot-pounds  of  energy  more  than  it  had  before.  This  is  often 
called  the  energy  of  the  clock-weight,  but  it  should  never  be 
forgotten  that  the  energy  really  resides  in  the  system,  earth-and- 
clock weight,  and  is  due  to  the  relative  separation  of  its  parts 
against  the  force  of  gravity  acting  between  them. 

Similarly,  the  water  in  a  lake  or  mill-dam  can  do  work  if 
allowed  to  fall  to  a  lower  level ;  to  the  amount  of  one  foot-pound 
for  every  pound  of  water  that  descends  a  vertical  height  of  one 
foot. 

Again,  work  must  be  done  to  wind  up  a  watch-spring,  or  bend 
a  bow.  The  coiled  watch-spring  and  the  bent  bow  possess  an 
equivalent  amount  of  energy,  and  will  do  the  work  back  again  if 
allowed  to  unbend.  In  these  cases  the  particles  on  the  inner 
side  of  the  spring  or  bow  have  been  forced  together,  and  those 
on  the  outer  side  drawn  apart,  against  the  elastic  forces  of 
cohesion  that  hold  the  bow  together.  When  the  constraint  is 
released,  these  forces  bring  the  particles  of  the  bow  back  to  their 
positions  of  equilibrium,  and  thus  do  upon  the  arrow  the  work  that 
was  stored  up  in  the  bow  by  the  force  used  in  bending  it.  The 
arrow  leaves  the  bow  with  the  corresponding  amount  of  kinetic 
energy,  which,  again,  it  can  give  up  only  on  meeting  some  obstacle, 
or  otherwise  experiencing  a  force  from  the  action  of  some  body  not 
moving  at  the  same  rate. 

142.  The   name  Potential  Energy  has   been   given  to  that 
form   of  energy,  or   capacity   for   doing  work,  which   is   due  to 
a  mutual  displacement  of  objects  or  parts  of  an  object  against  the 
forces  which  hold  them  together.     It  is  quite  possible  that  all 
forms  of  energy,  including  those  now  classed  as  Potential,  may 
ultimately  be  reduced  to  cases  of  kinetic  energy.     Meanwhile  the 
word  Potential  must  not   be    misunderstood   to   mean  that  this 
form  of  energy  is  not  really  energy  at  all,  but  only  something 
which  may  become  energy,  if  allowed  to  convert  itself  into  the 
kinetic  form. 

143.  No  formula  can  be  given  for  the  calculation  of  Potential 
energy,  so  universally  applicable  as  the  expression  M F2/2  for  the 


XV]  WORK  AND   ENERGY  147 

energy  of  a  moving  body.  But  very  often  it  can  be  reckoned 
easily.  The  potential  energy  of  raised  weights  is  at  once  ex- 
pressed in  foot-pounds  by  multiplying  the  number  of  pounds  by 
the  height  in  feet  through  which  they  are  to  fall.  In  the  case  of 
elastic  bodies,  we  often  know  the  force  which  was  employed  in 
producing  the  distortion.  Thus  if  we  know  the  average  force 
used  in  drawing  the  bow,  and  the  length  of  the  arrow  drawn,  the 
product  gives  the  energy  stored  in  the  bent  bow. 

144.  Wherever  bodies  are  in  motion  under  the  action  of 
forces,  work  is  being  done,  and  equivalent  amounts  of  energy  stored 
in  the  system  or  expended.  Mechanical  processes,  indeed,  may  be 
regarded  as  cases  of  the  transfer  of  energy  from  one  body  or  system 
to  another,  or  transformations  from  one  of  its  forms  to  another. 
Every  such  transfer  involves  the  exertion  of  a  force  while  the  body 
moves  over  a  certain  distance.  We  may  at  our  pleasure  either 
(1)  regard  the  transfer  of  the  energy  as  the  result  of  the  work 
done  by  the  force,  and  measure  it  by  the  product  of  the  force 
into  the  distance  moved,  P  x  s ;  or  (2)  consider  the  force  a 
manifestation  of  the  transfer  of  energy.  From  this  point  of  view 
the  force  is  measured  by  the  quotient  of  the  energy  transferred,  or 
work  done,  divided  by  the  distance  moved,  i.e.  the  force  is  the 
space-rate  of  transfer  of  energy. 

(1)     In   drawing   a   bow  the   archer   exerts   a   force   on   the 
arrow  and  draws  it  through  its  length.     He  does 

Illustrations  of  .  ° 

the  Transfer  of  work,  which  is  stored  as  potential  energy  in  the 
bent  bow,  in  virtue  of  the  relative  displacement 
of  its  parts.  When  the  arrow  is  released,  the  bow  does  this  work 
on  the  arrow,  exerting  a  force  on  it  by  means  of  the  string,  while 
the  arrow  moves  its  length.  The  energy  is  thus  transferred  to 
the  arrow,  and  at  the  same  time  transformed  from  potential  to 
kinetic  energy,  due  to  the  motion  of  the  arrow  relative  to  the 
earth,  including  the  bow  and  other  surrounding  objects.  The 
energy  cannot  be  transferred  again  until  the  arrow  meets  some 
object,  relatively  at  rest,  which  can  exert  a  force  upon  it;  but  if 
the  arrow  is  rising  against  gravity,  part  of  its  kinetic  energy  will 
be  transformed  into  potential  energy;  so  that  at  every  moment 
of  its  flight  it  possesses,  in  place  of  the  kinetic  energy  it  has  lost, 

10—2 


148  MECHANICS  [CHAP 

an  equivalent  amount  of  potential  energy,  due  to  its  separation 
from  the  earth  against  its  weight 

(2)  In  the  common  pendulum  the  transformation  of  energy 
from  the  potential  to  the  kinetic  form,  and  back  again,  takes  place  at 
every  swing.    The  energy  is  first  stored  in  the  pendulum,  when  it  is 
drawn  aside  by  the  exertion  of  force,  as  potential  energy  measured 
by  the  product  of  the  weight  of  the  bob  into  the  vertical  height 
above   the   lowest   position.     During   the   descent  the   potential 
energy  is  gradually  converted  into  equivalent  energy  of  motion, 
and  has  become  entirely  kinetic  at  the  bottom  of  the  swing.     At 
each  point  of  the  descent  the  sum  of  the  kinetic  and  potential 
energies  is  the  same.     During  the  ascent  the  kinetic  energy  is 
expended  as  the  pendulum  climbs  against  gravity,  and  when  it 
reaches  the  same  height  on  the  other  side,  the  whole  of  it  has 
again  been  converted  into  potential  energy,  and  the  swing  re- 
commences. 

This  process  might  be  repeated  indefinitely  but  for  frictions 
and  resistances,  hitherto  left  out  of  consideration,  which  absorb 
a  small  quantity  of  energy  in  each  swing.  It  is  the  business  of 
the  descending  clock-weight  to  supply  this  small  loss  by  means  of 
the  escapement,  and  so  maintain  the  swing.  The  energy  of  the 
clock-weight,  again,  is  supplied  by  the  work  done  in  winding  it  at 
intervals. 

(3)  A  similar  instance  in  which  it  is  easier  to  trace  the  force 
exerted    during   the   transfer   of  energy,   is    found   in   At  wood's 
Machine  (§  131),  when  the  two  equal  weights  A,  B  are  allowed  to 
run  alone,  and  therefore  uniformly.     No  change  is  taking  place  in 
the  kinetic  energy  of  either  weight.     But  the  potential  energy  of 
the  descending  weight  A,  with  regard  to  the  earth,  is  decreasing ; 
and  that  of  the  ascending  weight  increasing  at  the  same  rate. 
The  force  effecting  the  transfer  is  the  tension  of  the  string,  which 
is  doing  work  against  the  one,  and  an  equal  amount  of  work  upon 
the  other. 

145.  Many  mechanical  processes  depend  upon  the  storing  of 
storage  of  energy  in  some  system,  by  doing  work  upon  it, 
Energy.  an(j  ^hen  aliowmg  the  system  to  give  up  the  work 

either  suddenly  or  slowly  according  to  convenience. 


XV]  WORK   AND    ENERGY  149 

Thus  the  clock-weight  or  watch-spring  is  wound  up  in  a  few 
seconds,  and  gives  out  its  stored  energy  slowly,  during  eight  days 
or  twenty-four  hours.  The  bow  gives  out  its  energy  almost 
instantaneously,  exerting  the  same  force  on  the  arrow  as  that 
used  by  the  archer  in  drawing  it,  but  following  up  the  arrow,  as 
its  speed  increases,  far  more  swiftly  than  the  archer  could  have 
done,  besides  enabling  him  to  do  the  hard  work  at  leisure,  and 
then  concentrate  his  attention  on  the  aim. 

The  Hammer  enables  us  to  exert  a  much  greater  force  than 
we  could  unaided.  Let  us  suppose  that  a  hammer-head  weighs 
one  pound,  and  that  we  draw  it  down  by  the  handle  with  a  force 
of  ten  pounds-weight,  through  a  vertical  height  of  two  feet,  on  to 
the  head  of  a  nail.  What  must  have  been  the  pressure  exerted 
on  the  nail,  if  we  find  it  is  driven  in  half  an  inch  by  the  blow  ? 

During  the  descent  both  the  pull  of  gravity  (1  pound)  and  the 
force  (10  pounds)  have  been  doing  work,  so  that  energy  has  been 
stored  in  the  hammer-head  to  the  extent  of 

11  x  2  foot-pounds. 

It  is  in  the  form  of  kinetic  energy  at  the  moment  of  striking ; 
and  assuming  that  it  is  all  taken  up  in  overcoming  the  resistance 
of  the  nail  (average  value  =  R),  while  the  nail  recedes  half  an  inch 
(  =  ^4  ft.),  we  have 

Rx  ^?  =  11x2. 

.'.     R  =  528  pounds- weight. 

This  is  the  resistance  required  to  exhaust  the  kinetic  energy 
of  the  hammer  in  half  an  inch.  But  the  weight  of  the  head 
continues  to  act  during  the  process,  and  if  the  force  of  10  pounds 
is  also  applied,  we  must  add  11  pounds  more,  so  that  the  total 
average  pressure  must  have  been  539  pounds.  This  is  the  dead 
weight  which  the  nail  could  just  support  without  being  driven  in. 

The  Punching  Machine  is  a  machine  for  punching  holes 
through  thick  plates  of  metal.  At  first  sight  it  is  difficult  to 
conceive  of  any  tool  being  driven  with  such  force  as  to  cut  a 
J-inch  hole  through  an  inch  plate  of  cold  steel  quickly  and  quietly. 
It  is  easily  done  as  follows.  The  tool  is  attached  to  the  short  end 
of  a  lever  whose  long  end  is  forced  up  by  a  cam,  or  projection  on 
a  wheel,  which  only  comes  round  once  in  every  seven  or  eight 


150  MECHANICS  [CHAP. 

revolutions  of  the  engine  which  drives  the  machine.  During  six 
or  seven  strokes  the  engine  does  work  on  a  heavy  fly-wheel,  and 
when  the  cam  comes  round,  the  whole  of  the  kinetic  energy  stored 
in  the  now  rapidly  revolving  wheel  is  brought  to  bear  on  the  tool. 
When  it  comes  in  contact  with  the  plate  one  of  three  things  must 
happen.  Either  (1)  this  energy  must  disappear,  the  machine 
being  suddenly  brought  to  rest  without  equivalent  work  done ; 
or  (2)  the  machine  must  break ;  or  (3)  the  plate  must  be  punched. 
But  the  laws  of  motion  will  certainly  not  fail ;  and  it  is  the 
business  of  the  manufacturer  to  make  the  machine  strong  enough 
not  to  break.  The  only  alternative  is  that  the  plate  must  be 
punched;  and  accordingly  it  is. 

146.  Heavy  fly-wheels  are  used  for  another  purpose,  to  ensure 
Regulation  of  the  the  steady  running  of  an  engine.  The  steam  does 
supply  of  Energy.  work  on  foe  piston  at  very  different  rates  at  dif- 
ferent parts  of  the  stroke ;  and  at  the  beginning  and  end  of  the 
stroke,  the  two  dead  points,  no  work  is  being  done  at  all.  If  the 
engine  were  coupled  directly  to  the  machinery  of  a  factory,  each 
machine  would  run  in  a  series  of  jerks;  and  should  one  or  two 
machines  be  disconnected  or  brought  into  action,  the  speed  of  all 
the  rest  would  be  suddenly  and  seriously  affected.  But  if  a  very 
heavy  fly-wheel  be  attached  to  the  shaft,  the  engine  pumps  energy 
into  it  at  a  rate  varying  throughout  the  stroke,  but  the  machines 
draw  off  their  supply  from  the  large  store  accumulated  in  the 
wheel.  The  energy  of  such  a  wheel  can  be  calculated  (§  260) 
when  its  mass,  dimensions,  and  speed  are  known.  It  is  then  easy 
to  design  a  wheel  whose  energy  at  the  normal  speed  shall  be  any 
number  of  times  the  whole  amount  supplied  by  the  engine  during 
one  stroke;  so  that  the  speed  cannot  vary  during  the  stroke  by 
more  than  a  small  fraction,  and  will  not  change  very  greatly  even 
if  a  number  of  extra  machines  be  suddenly  thrown  in  or  out  of 
gear.  The  water  supply  of  a  large  town  is  managed  on  precisely 
the  same  principle.  The  engines  are  not  connected  directly  to 
the  service  pipes,  or  the  water  would  issue  in  sudden  jets ;  but  the 
pumps  work  into  a  large  reservoir,  from  which  the  supply  is  drawn 
off.  No  appreciable  change  is  caused  in  the  level  of  the  reservoir 
(and  therefore  in  the  steady  pressure  of  the  service)  either  by  the 


XV]  WORK  AND   ENERGY  151 

intermittent  strokes  of  the  pumps,  or  by  the  casual  turning  on  or 
off  of  taps  in  other  parts  of  the  city. 

147.     In  all  the  cases  so  far  considered  there  has  been  no  gain 
or  loss  of  energy  on  the  whole,  but  only  a  transfer 

Reversible  and 

irreversible  from  system  to  system,  or  from  one  form  to  an- 

other. What  has  been  gained  or  lost  in  one  shape 
has  been  lost  or  gained  in  another.  The  work  done  in  winding 
up  the  clock- weight  can  be  recovered  by  letting  it  run  down ;  the 
bent  bow  restores  in  unbending  the  work  required  to  bend  it ;  the 
pendulum  rises  to  an  equal  height  on  the  other  side  of  the  vertical. 

Mechanical  processes  of  this  kind,  which  can  be  run  backwards 
with  recovery  of  the  whole  of  the  original  work,  are  called  reversible. 

In  practice  they  are  generally  accompanied  by  others  which 
are  irreversible.  The  Simple  Machines  and  their  combinations 
do  not  give  the  results  demanded  by  the  formulae  we  have  ob- 
tained. Their  movements  are  interfered  with  by  frictions  and 
resistances,  wherever  their  moving  parts  come  into  contact  with 
each  other  or  with  the  surrounding  air.  Even  the  pendulum  is 
affected  by  resistances  at  the  pivot  and  against  the  air. 

These  forces,  being  only  called  forth  by  motion,  by  their  very 
nature  always  act  so  as  to  oppose  it.  When  therefore  the  machine 
is  run  backwards,  their  direction  is  reversed,  and  instead  of  the 
work  expended  against  them  being  restored,  more  is  used  up. 
It  is  the  business  of  Physics  to  trace  what  becomes  of  the  energy 
which  thus  passes  away  from  the  ken  of  Mechanics.  Here  it  may 
only  be  said  that  when  account  is  taken  of  all  the  other  effects 
accompanying  mechanical  processes, — the  heat,  the  sounds,  the 
luminous,  electric,  magnetic,  and  chemical  changes, — it  is  found 
that  the  total  energy  of  a  system,  isolated  and  left  entirely  to 
itself,  though  it  may  take  on  many  forms,  is  unalterable  in  amount. 
This  is  the  doctrine  of  the  Conservation  of  Energy,  the  central 
landmark  of  the  science  of  the  nineteenth  century.  The  principle 
began  to  be  clearly  apprehended  from  1840  onwards,  first  in  the 
case  of  Heat  through  the  work  of  Joule  and  Mayer ;  and  was  first 
formally  extended  to  all  branches  of  Physics  in  1847  by  Helmholtz 
in  his  paper  on  u  Die  Erhaltung  der  Kraft."  Maxwell  gives  it  the 
following  general  statement : 


152  MECHANICS  [CHAP. 

"  The  total  energy  of  any  body  or  system  of  bodies  is  a  quantity 
which  can  neither  be  increased  nor  diminished  by  any  mutual  action 
of  these  bodies,  though  it  may  be  transformed  into  any  of  the  forms 
of  which  energy  is  susceptible." 

148.     The    Mechanical    Powers    and    their    combinations    are 

incapable  of  producing  a  supply  of  work.     They 

En"rgyS°fThe          can  onty  transfer  or  transform  an  existing  supply, 


so  as  to  apply  the  force  in  some  specially  con- 
venient way.  They  may  increase  the  force,  but 
in  this  case  the  distance  through  which  it  moves  the  body  is 
decreased  in  the  same  proportion,  or  decrease  the  force,  but  in  this 
case  the  distance  is  increased  in  the  same  proportion.  "  What  is 
gained  in  power  is  lost  in  speed."  Machinery  cannot  produce 
work  for  us;  it  has  to  be  worked. 

The  energy  needed  for  driving  a  machine  must  be  obtained 
from  such  sources  as  living  animals,  the  kinetic  energy  of  winds, 
the  potential  energy  of  water-falls,  the  chemical  energy  stored  in 
coal  or  other  fuel.  From  these  must  be  supplied  not  only  the 
useful  work  delivered  by  the  machine,  but  the  waste  and  loss 
due  to  irreversible  processes  in  working  it. 

149.  The  Power  of  any  of  these  agents  is  measured  by  the 
rate  at  which  it  can  supply  work.     Many  years  ago  James  Watt 
made  experiments  on  the  power  of  some  of  the  heavy  dray-horses 
belonging  to  Barclay  and  Perkins'  Brewery,  London.     The  horses 
were  set  to  raise  a  weight  of  100  Ibs.  from  the  bottom  of  a  deep 
well   by  pulling   horizontally  on  a  rope  passing  over  a  pulley. 
Watt  found  that  a  horse  could  walk  about  2J  miles  an  hour  at 
this  work,  thus  doing  2'5  x  5280  x  -V°o°-  =  22,000   foot-pounds  per 
minute.     Allowing  50  °/0  extra  for  the  work  wasted  in  frictions,  he 
arrived  at  the  estimate  of  33,000  foot-pounds  per  minute  for  the 
average  power  of  a  horse.   This  unit  has  been  adopted  by  engineers, 
and  is  known  as  a  Horse-Power. 

150.  When  several  forces  act  on  a  moving  body,  each  does 
The  case  of  work,  or  has  work  done  against  it.     If  the  work 
Equilibrium.  done  by  some  of  them  is  equal  to  the  work  done 
against  the  others,  the  kinetic  energy  of  the  body  is  unchanged. 


XV]  WORK   AND    ENERGY  153 

But  uniform  motion  can  only  take  place  when  on  the  whole  no 
forces  are  acting.  The  forces  in  this  case  must  therefore  balance, 
or  be  in  equilibrium.  This  is,  in  fact,  the  Principle  of  Virtual 
Work,  already  treated  in  Chapter  VII. 

Thus  when  the  equal  weights  of  an  At  wood's  Machine  (§  131) 
are  running  uniformly,  the  upward  tension  of  the  string  on  either 
weight  is  exactly  equal  and  opposite  to  the  downward  pull  of 
gravity.  The  forces  are  in  equilibrium,  just  as  they  would  be  in 
the  special  case  when  the  velocity  of  the  system  was  zero,  i.e. 
when  it  was  at  rest  relatively  to  the  earth. 

We  shall  now  consider  the  equilibrium  of  forces  from  the 
Newtonian  point  of  view  in  more  detail. 


EXAMPLES. 

(Caution,  In  using  the  dynamical  formulae  MV=Pt;  MV^/2  =  Ps,  the 
force  must  always  be  expressed  in  the  proper  dynamical  units,  i.e.  poundals 
or  dynes,  according  to  the  system  employed  ;  conversely,  forces  determined 
from  these  formulae  must  be  converted  to  pound-  or  gramme-weights  by 
dividing  by  32 '2  or  981  respectively.) 

1.     Calculate  in  foot-pounds  the  energy  of : 

(a)     a  projectile  weighing  1034  Ibs.,  and  having  a  velocity  of  2262  ft. 
per  second ; 

(6)     a  train  of  300  tons  moving  at  60  miles  an  hour. 

2  The  projectile  in  Question  1  penetrates  a  sandbank  to  a  depth  of 
30  ft. !  the  train  is  brought  to  rest  by  the  brakes  in  one  minute.  Compare 
the  resistance  offered  by  the  sandbank  with  the  retarding  force  of  the  brakes. 

3.  The  unit  of  Power  commonly  employed  in  electrical  engineering  is 
that  of  the  c.G.s.  system,  the  Watt,  which  is  a  power  capable  of  doing  107 
ergs  of  work  per  second.     If  1  lb.  =  454  gms.,  1  metre  =  39 '37  inches,  and 
#  =  981,  shew  that  one  horse-power =746  watts. 

4.  A  man  weighing  12  stone  climbs  a  mountain  at  the  rate  of  1000  feet 
(vertically)  an  hour.     What  horse-power  is  he  developing? 

5.  What  must  be  the  horse-power  of  an  engine  to  pump  1100  cubic  feet 
of  water  per  hour  from  a  well  120  feet  deep,  a  cubic  foot  of  water  weighing 
1000  ounces? 

6.  A  train  weighs  120  tons  including  the  engine.     The  resistances  to 
motion  on  a  level  are  equivalent  to  a  retarding  force  of  16  Ibs.  weight  per 


154  MECHANICS  [CHAP.  XV 

ton.     Find  the  greatest  speed  at  which  the  train  can  run  if  the  engine  is  of 
150  H.P. 

("  Full  speed  "  is  the  speed  at  which  the  engine  is  just  able  to  exert  a  force 
equal  to  the  resistances  to  motion.  The  train  then  moves  uniformly  under 
the  First  Law  of  Motion.) 

7.  If  the  train  in  Question  6  is  moving  at  20  miles  an  hour,  and  the 
engine  is  working  at  full  power,  find  the  acceleration. 

8.  Find  the  H.P.  of  an  engine  which  can  take  a  train  of  100  tons  up  an 
incline  of  1  in  200  at  20  miles  an  hour,  the  resistances  being  equivalent  to 
14  Ibs.  per  ton.     (Here  the  engine  has  to  do  the  work  required  to  lift  the 
train  vertically  through  a  certain  height  per  minute,  as  well  as  to  overcome 
the  resistances.) 

9.  The  resistances  to  motion  of  a  train  being  14  Ibs.  per  ton  (English) 
weight,  if  the  train  going  40  miles  per  hour  come  to  the  foot  of  an  incline 
of  1  in  168,  the  steam  being  turned  off,  find  how  far  it  will  run  up  the 
incline. 

If  it  had  come  to  the  top  of  the  incline,  how  far  would  it  have  descended 
before  stopping  ? 

10.  A  bullet  weighing  half  an  ounce  is  fired  with  a  speed  of  2000  feet 
per  second  from  a  rifle  weighing  10  Ibs.     If  the  rifle  kicks  back  through 
3  inches,  find  the  average  pressure  applied  by  the  shoulder  in  bringing  it 
to  rest. 

11.  A  hammer-head  weighing  1  Ib.  strikes  a  nail  with  a  velocity  of  10 
feet  per  second,  and  drives  it  in  1  inch.     What  was  the  average  pressure  of 
the  hammer  on  the  nail  ? 

12.  The  ram  of  a  pile-driver  weighs  200  Ibs.  and  falls  12  feet  on  the 
head  of  a  pile  which  yields  half  an  inch.     What  steady  weight  could  the  pile 
sustain  ? 

13.  A  clock-weight  of  4  kilogrammes  is  wound  up  through  a  height  of 
1  metre  and  then  drives  the  clock  for  eight  days.     Express  in  Watts  the 
power  needed  to  drive  the  clock. 

14.  In  a  steam  engine  the  average  pressure  of  steam  during  the  stroke  is 
180  Ibs.  on  the  square  inch.     The  length  of  stroke  is  3  ft.  4  in.,  and  the 
diameter  of  the  piston  is  5^  inches.     If  the  engine  makes  125  revolutions  per 
minute,  find  its  horse-power. 

15.  An  ocean  steamer  with  engines  of  30,000  H.P.  can  make  25  miles  an 
hour.     What  is  the  resistance  to  her  motion  through  the  water  ? 

16.  The  average  flow  over  Niagara  Falls  is  270,000  cubic  feet  per  second. 
The  height  of  fall  is  161  feet.     What  horse-power  could  be  developed  from 
the  Falls  if  all  the  energy  were  utilized  ? 

17.  A  belt  is  transmitting  12  H.P.  to  a  pulley  2  feet  in  diameter,  running 
at  375  revolutions  per  minute.     What  is  the  driving  force  of  the  belt  ? 


» 


CHAPTER  XVI. 

THE   PARALLELOGRAM  LAW. 

151.  ACCOEDING  to  the  Second  Law  of  Motion,  when  two  or 
more  forces  act  at  a  point  of  a  body,  each  produces  its  effect 
independently  of  the  others,  and  this  effect  is  not  only  pro- 
portional to  the  magnitude,  but  takes  place  in  the  direction  of 
the  force. 

Forces  may  therefore  conveniently  be  represented  by  straight 
lines.     For  a  force  is  completely  specified   when 

Representation  of  *•  *        A 

Forces  by  straight  we  know  (1)  the  point  at  which  it  acts,  (2)  its 
direction,  and  (3)  its  magnitude ;  and  a  straight 
line  can  be  drawn  (1)  from  any  point,  (2)  in  any  direction,  and 
(3)  of  such  a  length  as  to  represent  any  magnitude  on  any 
convenient  scale. 

Quantities  which,  like  forces,  depend  for  their  effect  on  their 
direction  as  well  as  on  their  magnitude,  are  distinguished  as 
vector  quantities,  while  quantities  which  have  only  magnitude, 
such  as  a  sum  of  money,  or  the  amount  of  corn  in  a  heap,  are 
called  scalar  quantities.  Scalar  quantities  are  added  by  ordinary 
arithmetic.  But  a  special  rule  is  required  for  adding,  or  rather 
compounding,  vector  quantities.  This  rule  is  the  Parallelogram 
Law,  already  stated  for  forces  in  §  38.  We  proceed  to  prove  it  in 
turn  for  Displacements,  Velocities,  Accelerations,  and  Forces. 


156 


MECHANICS 


[CHAP. 


I.   Displacements. 


152.  Let  a  point  0  receive  two  separate  displacements 
represented  by  OA, 
OB  respectively.  The 
order  in  which  the  displacements  are 
given  is  immaterial.  We  may  suppose 
the  point  first  carried  to  A,  and  then 
displaced  through  AC,  equal  and 
parallel  to  OB ;  or  first  carried  to  B, 


Fig.  62. 


and  then  displaced  through  BC,  equal  and  parallel  to  OA.  The 
joint  result  is  the  same.  The  point  arrives  at  C,  which  it  might 
have  reached  by  a  single  displacement  represented  by  OC. 

A  single  displacement  can  thus  be  found  which  is  equivalent 
to  (i.e.  has  the  same  effect  as)  any  two  displacements;  it  is 
represented  by  that  diagonal  of  the  parallelogram  constructed 
on  the  lines  representing  the  displacements  which  passes  through 
the  point. 


II.    Velocities. 


153.  If  the  two  displacements  OA,  OB,  take  place  uniformly 
and  simultaneously  in  the  course  of  one  second, 
OA,  OB  will  represent  velocities,  and  OC  the 
single  velocity  which  is  equivalent  to  them.  The  rule  is  thus 
true  for  velocities. 

If  any  difficulty  is  found  in  conceiving  that  a  point  may  have 
two  velocities  at  once,  think  of  a  fly  crawling  along  the  paper 
along  OA  in  one  second,  while  the  paper  itself  is  moved  obliquely 
along  OB.  The  velocity  of  the  fly  with  regard  to  the  table  is  OC, 
which  may  be  regarded  as  made  up  of  his  velocity  with  regard  to 
the  paper,  OA,  together  with  that  of  the  paper  with  regard  to  the 
table,  OB. 

If  the   motions   take   place   uniformly,  by  the  time  the  fly 
reaches  any  point  A',  the  paper  will   . 
have  moved  a  proportional  distance  B, 

AC'  =  OB'  such  that 

AfC'  _AC_ 

OA'  ~  OA ' 

so  that  C'  is  on  OC,  and,  in  one 
second,  he  actually  moves  along  0(7, 
relatively  to  the  table. 


XVI]  THE   PARALLELOGRAM    LAW  157 

154.     (1)     A  steamer  steering  due  East  at  10  knots  an  hour 
,.     .  is  carried  by  a  current  due  North  at  3  knots  an 

Applications. 

hour.     Find  the  real  speed  and  course. 
The  speed  (Fig.  64)  is 

OG  =  VlO2  +  32  =  V109  =  10-44, 
0=16°  42'  north  of  East. 


»>-iE 


w 


E 


Fig.  65. 


(2)  A  vessel  makes  6  knots  an  hour  due  West.  Another  is 
making  8  knots  due  South.  What  is  the  speed  and  course  of  the 
second  with  regard  to  the  first  ? 

Cases  of  relative  motion,  such  as  this,  are  best  solved  by  the 
following  artifice.  No  difference  will  be  produced  in  the  relative 
motion,  if  each  of  the  moving  objects  is  given  an  extra  velocity, 
provided  it  is  the  same  for  each. 

Let  J.,  B  (Fig.  65)  be  the  vessels.  Apply  to  each  the  velocity  6 
knots  due  East,  which  is  equal  and  opposite  to  the  actual  velocity  of 
A.  The  effect  will  be  that  A  is  reduced  to  rest,  having  equal  and 
opposite  velocities;  while  B  moves  with  the  two  speeds  8  knots 
South  and  6  knots  East  jointly.  But  these  are  equivalent  to  a 
speed  V62  +  82  =  10  knots,  at  an  angle  tan"1  3/4  east  of  South. 
This  is  the  speed  and  course  relative  to  A  supposed  at  rest. 

If  the  position  of  B  with  regard  to  A  is  given,  it  is  easy  to 
calculate  whether  there  will  be  a  collision,  or  what  will  be  the 
shortest  distance  between  the  ships. 


158  MECHANICS  [CHAP,  xvi 

155.  If  the  velocities  OA,  OB  are  communicated  to  a  point 

during  one  second,  it  has  accelerations  OA,  OB: 

III.  Accelerations.  .°  .  .  .  ' 

but  it  is  obvious  that  the  effect  is  the  same  as  if 
the  equivalent  velocity  OC  were  imparted  every  second.  That  is, 
an  acceleration  OC  is  equivalent  to  the  two  accelerations  OA,  OB. 

156.  Let   two   forces   represented   by   OA,  OB  act   on   the 

same   mass.     By   the 

IV.  Forces.  *•»»,• 

Second  Law  ot  Motion 
they  will  produce,  independently,  ac- 
celerations in  the  directions  OA,  OB 
and  proportional  to  them.  OA,  OB 
may  therefore  be  taken  to  represent 
the  accelerations. 

But  a  single  acceleration  OC  is  equivalent  to  OA,  OB  jointly; 
and  by  the  second  law  this  might  have  been  produced  by  a  single 
force  acting  in  the  direction  OC,  and  represented  by  OC  on  the 
same  scale  as  that  on  which  the  original  forces  are  represented 
by  OA,  OB. 

Hence  the  force  OC  is  equivalent  to  the  two  forces  OA,  OB} 
and  the  rule  is  true  for  forces. 


EXAMPLES. 

1.  A  train  is  travelling  due  North  at  20  miles  an  hour  through  a  shower 
of  rain  falling   almost  vertically,  but  with  a  slight  in cli nation   eastwards, 
enough  to  make  the  drops  graze  the  windows.     If  the  raindrops  have  a  speed 
of  16  feet  per  second,  find  the  inclination  to  the  vertical  of  the  splashes  on 
the  windows. 

2.  A  shot  with  a  velocity  of  2000  feet  per  second  is  fired  at  a  steamer  in 
a  direction  at  right  angles  to  the  steamer's  course,  and  pierces  both  sides. 
If  the  deck  is  40  feet  broad,  and  the  steamer  is  making  25  miles  an  hour, 
find  how  many  inches  the  second  hole  will  be  astern  of  the  first. 

3.  The  speed  of  the  earth  in  her  orbit  is  19  miles  per  second.     Con- 
sequently the  light  from  a  star  appears  to  be  slightly  altered  in  direction  to 
an  observer  on  the  earth,  and  the  star  is  apparently  displaced  in  the  direction 
of  the  earth's  motion.     This  "aberration"  from  the  true  position  (discovered 
by  Bradley  in  1729)  is  20 '45"  for  a  star  situated  in  a  direction  perpendicular 
to  the  earth's  line  of  motion      Hence  find  the  velocity  of  light. 


CHAPTEE  XVII. 

THE  COMPOSITION  AND   EESOLUTION  OF  FORCES. 
RESULTANT.    COMPONENT.    EQUILIBRIUM. 

157.  BY  means  of  the  Parallelogram  of  Forces  two  or  more 
forces  acting  at  a  point  may  be  compounded  into  a  single  force, 
called  their  Resultant,  which  shall  produce  the  same  effect. 
And  this  effect  can  in  general  be  more  easily  calculated  for  the 
single  resultant  than  for  the  several  forces  to  which  it  is 
equivalent. 

In  particular,  when  there  is  to  be  no  change  at  all  in  the  state 
of  rest  or  motion,  i.e.  when  there  is  to  be  equilibrium,  the  resultant 
must,  by  the  Second  Law  of  Motion,  be  zero.  Any  mathematical 
expression  of  this  fact  is  a  statement  of  the  conditions  of  equi- 
librium for  the  given  forces. 

Conversely,  a  single  force  may  be  resolved  into  two  or  more 
Components,  which  shall  together  have  the  same  effect.  This  is 
often  convenient  especially  when  we  wish  to  limit  our  attention 
to  the  motion  or  conditions  of  equilibrium  in  a  particular 
direction ;  for  each  force  can  be  resolved  into  a  component  along 
that  direction,  and  another  perpendicular  to  it ;  and  the  latter 
may  be  disregarded,  as  it  can  produce  no  effect  at  right  angles  to 
its  own  line  of  action. 

The  magnitudes  and  directions  of  the  straight  lines  repre- 
senting the  Resultants  and  Components  are  to  be  found  by 
geometrical  construction  or  trigonometrical  computation. 


160 


MECHANICS 


[CHAP. 


Forces  acting  in 
the  same  plane  at 
the  same  point. 


158.     I.     To  find  the  Resultant  of  Two  Forces. 
(a)     Geometrical  Methods. 
(1)     Construct   the    parallelogram    and    draw 
the  diagonal. 

(2)     The   whole    parallelogram   need   not   be   drawn.      Take 

OA  to  represent  the  first  force,  and  B  c 

from  A   draw   AC  representing   the 

second   in   magnitude   and   direction 

(but  not  in  point  of  application). 

Join  0(7.     This  is  the  Resultant. 

It  is  better,  especially  when  there 

are  several  forces,  to  make  two  figures : 


Fig.  67. 


a  force  diagram,  where  the  lines  represent  the  forces  completely, 
and  a  construction  diagram  of  the  triangles  giving  the  magnitudes 
and  directions.  Thus : 


Fig.  68. 


(3)  Since  the  diagonals  bisect  each  other,  OC—20D,  where 
D  is  the  middle  point  of  AB.  This  value  of  the  resultant  is 
occasionally  convenient. 

(b)     Trigonometrical  Method. 

Let  the  two  forces  be  P  and  Q,  inclined  at  an  angle  a ;  let  R 
be  their  resultant,  making  an  angle  0  with  P. 

Then         0<72  =  OA*  +  AC*  -  WA  .  AC  cos  OAC, 
/.  R*  =  P>+Q>-  2PQ  cos  (180°  -  a) 
=  P2  +  Q2  +  2PQ  cos  a. 

.     ,  sin  6     sin  A 

And  -     =  - 


XVll]          THE   COMPOSITION   AND   RESOLUTION   OF   FORCES  161 


/.  sin  0  =      sin  (180°  -a) 
0    . 


(Caution.  If  other  forces  have  to  be  combined  with  the 
resultant  of  these  two,  the  whole  work  has  to  be  done  over  again 
for  each  force,  and  the  expressions  become  very  cumbrous.  The 
method  must  never  be  employed  for 
three  or  more  forces,  though  it  is  Q 
occasionally  convenient  when  there 
are  only  two.  For  more  than  two 
the  method  of  §  164  must  be  used.) 

The  special  case  when  the  forces 
are  at  right  angles  is  important.  i 

Here       £2  =  P2+Q2, 
and  tan  6  =  -^ . 


Fig.  69. 


159.     II.     To  resolve  a  Force  into  two  Components. 

This  can  be  done  in  an  infinite  number  of  ways.  For  draw 
any  triangle  on  the  line  representing  the  force  (Fig.  70).  The 
sides  give  the  magnitudes  and  directions  of  two  components  equi- 
valent to  the  force.  Thus : 


Fig.  70. 


Fig.  71. 


Observe  that  the  sides  of  the  triangle  must  be  taken  in  order, 
i.e.  we  must  continue  along  them  in  the  same  direction  round  the 
triangle.  The  resultant  of  OP  and  RP,  applied  at  0,  is  not  OR 
but  OR  (Fig.  71). 

11 


c. 


162 


MECHANICS 


[CHAP. 


160.  (1)   To  resolve  a  force  into  two  components  of  given 
magnitudes. 

This  is  to  construct  a  triangle  when  the  three  sides  are  given 
(Euc.  I.  22).  Evidently  the  two  components  must  together  be 
greater  than  the  force,  or  there  is  no  solution. 

161.  (2)   To  resolve  a  force  into  two  components  in  given 
directions. 

Let  OR  represent  the  force ;  and  let  Ox,  Oy  be  the  directions. 
Draw  parallels  to  the  directions  through  R.     OP,  OQ  are  the 
components  (Fig.  72). 


p 
Fig.  72. 


Fig.  73. 


162.  Let  the  force  OR  and  the  direction  Ox  be  fixed,  while 
Oy  varies.  Then  for  every  different  direction  given  to  Oy  the 
component  OP  along  Ox  has  a  different  value.  The  special  case 
when  OQ  is  at  right  angles  to  OP  is  so  important  that  the  value 
of  the  component  OP  in  that  case  is  called  The  Resolved  Part  of 
OR  in  the  direction  Ox,  the  corresponding  value  of  OQ  being 
The  Resolved  Part  of  OR  in  the  perpendicular  direction. 

Let  X,  Y  be  the  resolved  parts  of  the  force  R,  represented  by 
OR  (Fig.  73),  along  Ox,  Oy,  and  let  R  make  the  angle  0  with  Ox. 

Then  X  =  OP  =  OR  cos  0  =  R  cos  0, 

and  Y  =  OQ  =  OR  sin  6  =  R  sin  (9. 

To  find  the  resolved  part  of  a  force  in  any  direction  multiply  it 
by  the  cosine  of  the  angle  between  the  force  and  that  direction. 

Since  sin  6  =  cos  (90°  -  6)  =  cos  QOR,  the  rule  just  stated 
applies  to  the  component  Y  as  well  as  to  the  component  X.  If 
the  component  in  any  direction  is  found  by  multiplying  by  the 


XVIl]          THE   COMPOSITION   AND   RESOLUTION   OF   FORCES  163 

cosine  or  sine  of  any  angle,  then  the  component  in  the  per- 
pendicular direction  is  found  by  multiplying  by  the  sine  or  cosine 
of  the  same  angle. 

The  reason  for  the  importance  of  this  case  is  easily  seen. 
Suppose  0  (Fig.  73)  to  be  a  curtain  ring  sliding  on  a  smooth  rod 
Ox,  and  pulled  obliquely  by  a  cord  with  a  force  R  along  OR.  The 
ring  can  only  slide  along  the  rod.  In  finding  whether  it  will 
remain  at  rest  or  begin  to  move  we  are  not  helped  by  resolving  R 
into  components  P  and  Q  as  in  Fig.  72,  for  then  besides  the 
component  P  along  the  rod,  the  oblique  force  Q  will  still  have  to 
be  reckoned  with.  But  if  R  be  replaced  by  a  component  along  the 
rod  and  another  at  right  angles  to  it,  the  latter  may  be  left  out  of 
account,  since  it  can  produce  no  effect  in  the  direction  of  the  rod. 
It  is  a  great  simplification  to  have  thus  got  rid  of  all  oblique 
forces. 

163.     III.     Three  or  more  forces, 
(a)     Geometrical  Method. 


Let  the  forces  P]}  P2,  P3, ...  act  at  0  as  in  Fig.  74. 

Make  a  construction  diagram.  From  any  point  0'  draw  O'A 
to  represent  Px;  from  A  draw  AB  to  represent  P2;  and  so  on. 
Let  DE  represent  the  last  force. 

Then  O'B  represents  the  resultant  of  P1  and  P2;  O'C  the 
resultant  of  O'B  and  P,,  i.e.  of  Plt  P2  and  Ps ;  and  finally  O'E  the 
resultant  of  all  the  forces. 

Draw  OR  equal  and  parallel  to  O'E.     This  is  the  resultant. 

11—2 


164  MECHANICS 

164.     (b)   Trigonometrical  Method. 


[CHAP. 


Fig.  75. 

Let  the  forces  Plt  Pz, ...  act  at  0. 

Choose  any  direction  xOx',  and  yOy'  at  right  angles  to  it.  Let 
the  forces  make  angles  alt  «2, ...  with  Ox. 

Kesolve  each  of  the  forces  Ply  P2,  ...  into  its  components 
along  Ox,  Oy. 

The  components  of  Pl  are  Pl  cos  a^  along  Ox  and  Pj  sin  aa  along  Oy, 
„    P2   „   P2coso2     „      Ox    „    P2sina2      „      Oy, 

and  so  on  for  all  the  forces. 

The  oblique  forces  are  thus  got  rid  of,  and  we  have  only  a  set 
of  forces  PT  cosa1}  P2cos  «2,  &c.  acting  in  the  same  direction  along 
Ox\  and  another  set  P1sinall  P2sina2,  &c.  along  Oy. 

Let  the  sum  of  the  forces  along  Ox  be  X ;  that  of  the  forces 
along  Oy  IOQ  T\  so  that 

X  =  Pi  COS  «j  +  P2  COS  «2  +  . . ., 

F=  P!  sin  «!  +  P2  sin  «2  4-  — 


XVII] 


THE   COMPOSITION   AND   RESOLUTION   OF   FORCES 


165 


Then  the  original  set  of  forces  is  reduced  to  two  forces  X  and  Y 
acting  at  right  angles. 


Y/ 


X-' 


IO 


Fig.  76. 


The  resultant  of  these  is  R,  where 


and  it  makes  an  angle  0  with  Ox,  such  that 


.(1), 

.(2). 


The  advantage  of  this  method  is  that  however  many  forces 
there  may  be,  X  and  Y  can  be  written  down  at  once,  as  the  values 
of  the  cosines  and  sines  are  found  from  the  tables.  Then  21  and 
B  are  easily  found  from  (1)  and  (2). 

165.     I.   Two  Forces. 

Two  forces  acting  at  a  point  can  only  balance,  i.e.  fail  to  have 
The  conditions  of  a  resultant,  when  they  are  equal  in  magnitude  and 
Equilibrium.  directly  opposed  to  each  other. 


166  MECHANICS  [CHAP. 

(a)  Graphically. 

For  it  is  only  when  the  above  conditions  are  fulfilled  that  the 
diagonal  of  the  parallelogram  vanishes. 

(b)  Analytically. 

The  Trigonometrical  formula  for  the  resultant  leads  to  the  same 
result,  as  follows  : 


This  can  only  vanish  if  P  —  Q  =  0,  and  cos  ^  =  0,  i.e.  if  P  = 
and  a  =  180°. 

166.     II.   Three  Forces 
(a)     Graphically. 


Fig.  77. 

Make  the  force  diagram  O'AB  for  Plt  P2 ;  their  resultant  is  O'B. 
In  order  that  this  may  balance  P3,  P3  must  be  represented  in 
magnitude  and  direction  by  BO'  (taken  in  the  sense  of  the  arrow). 

Or,  make  the  force  diagram  O'ABG  for  all  three  forces.  Then 
unless  C  falls  upon  0',  they  will  have  a  resultant  O'C. 

Hence  :  In  order  that  three  forces  acting  at  a  point  may  be  in 
equilibrium  they  must  be  represented  in  magnitude  and  direction  by 
the  three  sides  of  a  triangle  taken  in  order. 

Conversely :    If  three  forces  represented  in  magnitude    and 


XVIl]          THE   COMPOSITION   AND   RESOLUTION   OF   FORCES  167 

direction  by  the  three  sides  of  a  triangle  taken  in  order  be  applied 
at  a  point,  they  will  be  in  equilibrium. 

This  proposition  is  known  as  the  Triangle  of  Forces. 

167.     (b)   Analytically. 

Father  Lami  in  his  Mdcanique  (published  in  1687,  the  year  of 
Newton's  Principia)  gave  the  Triangle  of  Forces  a  Trigonometrical 
form.  Produce  O'A,  AB,  BO'  (Fig.  77). 

Then  Z  P^OP*  =  Z  aAB  =  180°  -  A. 

Similarly,  Z  P2OP3  =  Z  bBO'  =  180°  -  B, 

Z  P3OP1  =  Z  o'O'A  =  180°  -  0'. 
For  equilibrium 

A   A-A 

0'A~AB     BO" 

A    Pi         P*         P* 
sin  B     sin  0'     sin  A  ' 

P!  P2  PS 


*  sinP2OP3     sinPsOP^sinP^P,,* 

Hence  Lami's  Theorem :  If  three  forces  acting  at  a  point  are  in 
equilibrium,  each  is  proportional  to  the  sine  of  the  angle  between  the 
other  two. 

168.  III.   Any  number  of  Forces. 
(a)     Graphically. 

Make  a  construction  diagram.  Then  there  will  be  a  resultant 
unless  the  last  point  returns  to  the  first,  and  the  diagram  forms  a 
closed  polygon. 

Hence :  If  any  number  of  forces  acting  at  a  point  are  represented 
in  magnitude  and  direction  by  the  sides  of  a  closed  polygon  taken  in 
order,  they  will  be  in  equilibrium. 

This  is  known  as  the  Polygon  of  Forces. 

169.  (b)   Analytically. 

By  the  method  of  §  164  the  resultant  R  is  given  by 


168  MECHANICS  [CHAP. 

where  X  =  Plcosa1  + , 

F=P1sina1  + 

For  equilibrium  R  =  0, 

.-.  Z2+F2  =  0. 

But  since  the  squares  are  necessarily  positive,  this  can  only  be 
the  case  when  X  and  Y  are  separately  zero. 

/.  X  =  0) 

and  F=OJ"' 

i.e.  The  sums  of  the  resolved  parts  of  the  forces  in  any  two  directions 
at  right  angles  must  be  separately  zero. 

We  are  at  liberty  to  choose  any  two  directions  at  our 
convenience,  for  in  finding  the  resultant  (§  164)  the  directions 
Ox,  Oy  were  taken  arbitrarily. 

Both  X  and  Y  must  be  zero.  If  JT  =  0,  there  can  be  no 
resultant  tending  to  cause  motion  along  Ox.  But  there  may  still 
be  an  unbalanced  force  along  Oy,  and  yet  no  effect  produced  along 
Ox,  at  right  angles  to  it.  It  is  necessary  therefore  to  have  F=0 
as  well. 

The  two  conditions  secure  that  there  shall  be  no  disturbance  in 
either  of  two  mutually  perpendicular  directions.  There  cannot 
then  be  an  unbalanced  force  in  any  other  direction,  since  had  such 
an  oblique  force  existed,  it  must  have  had  components  along  both 
Ox  and  Oy 

Note  that  (6)  can  be  at  once  deduced  from  (a)  by  projecting 
the  construction  diagram  on  to  any  straight  line  in  the  plane.  For 
the  sum  of  the  projections  of  the  sides  of  a  closed  polygon  on  any 
straight  line  is  zero. 

Conversely,  if  we  project  on  each  of  two  straight  lines,  and  find 
the  sum  of  the  projections  in  each  case  zero,  the  polygon  must  be 
closed. 


XVII]         THE   COMPOSITION   AND   RESOLUTION  OF  FORCES  169 


EXAMPLES. 

1.  Shew,  by  a  drawing,  that  if  the  angle  at  which  two  forces  are  inclined 
to  each  other  be  increased  their  resultant  is  diminished. 

2.  Hence  shew  that  if  a  picture  is  hung  from  a  nail  by  a  string  fastened 
to  two  rings  in  the  top  of  the  frame,  the  shorter  the  string  the  stronger  it 
ought  to  be.     Could  the  string  be  stretched  so  tightly  between  the  rings  as  to 
remain  straight  when  placed  over  the  nail  ? 

3.  Two  forces  acting  at  right  angles  to  each  other  have  a  resultant  which 
is  double  the  smaller  force.     Find  its  direction. 

4.  A  BCD  is  a  parallelogram,  and  AB  is  bisected  in  E\  prove  that  the 
resultant  of  the  forces  AD,  AC  is  double  the  resultant  of  EA,  AC. 

5.  A  BCD  is  a  quadrilateral,  and  JE'the  point  of  intersection  of  the  lines 
joining  the  middle  points  of  opposite  sides  ;  0  is  any  point.     Prove  that  the 
resultant  of  forces  OA,  OB,  OC,  OD  is  equal  to  40E. 

6.  Two  forces  act  at  a  point.     Shew  that  if,  when  one  of  the  forces  is 
reversed,  the  resultant  is  at  right  angles  to  the  direction  of  the  resultant 
before  the  change,  the  forces  are  equal. 

7.  A  BCD  is  a  quadrilateral.     Shew  that  if  four  forces  represented  by 
AB,  AD,  CB,  CD  be  applied  at  a  point,  their  resultant  will  be  represented  by 
four  times  the  line  joining  the  middle  points  of  the  diagonals. 

8.  Find  the  magnitude  and  direction  of  the  resultants  of  the  following 
pairs  of  forces  (in  pound  weights) : 

(1)  24  Ibs.  and  7  Ibs.  acting  at  right  angles, 

(2)  7  Ibs.  and  8  Ibs.  at  an  angle  of  60°, 

(3)  11  Ibs.  and  14  Ibs.  at  120°, 

(4)  6  Ibs.  and  8  Ibs.  at  52°. 

9.  Forces  7,  12,  3,  11  act  at  a  point,  the  first  due  East ;   the  second 
North-East,  the  third  North  ;  and  the  fourth  60°  west  of  North.     Find  their 
resultant. 

10.  Forces  1,  2,  3,  4,  5,  6,  7,  8  act  at  a  point,  the  angle  between  each 
force  and  the  next  being  47°.     Find  the  magnitude  of  the  resultant,  correct 
to  two  places  of  decimals,  and  its  direction. 

11.  Find  the  resolved  part  of  a  force  of  60  Ibs.  in  a  direction  inclined  40° 
to  the  force. 


170  MECHANICS  [CHAP. 

12.  A  canal-boat  is  pulled  by  a  rope  60  feet  long,  and  the  boat  is  30  feet 
from  the  towing  path.     If  the  horse  pulls  with  a  force  of  120  Ibs.  weight,  what 
is  the  force  urging  the  boat  forward  ? 

13.  A  captive  balloon  capable  of  raising  a  weight  of  400  Ibs.  is  anchored 
at  a  height  of  400  feet  by  a  rope  500  feet  long.     Find  the  strain  on  the  rope 
and  the  horizontal  pressure  of  the  wind  on  the  balloon. 

14.  A  50  Ib.  weight  hangs  by  a  wire  13  feet  long.     What  horizontal  force 
is  required  to  draw  it  aside  5  feet  from  the  vertical  through  the  point  of 
suspension,  and  what  will  then  be  the  tension  of  the  wire  1 

15.  A  body  of  weight  15  Ibs.  is  placed  on  an  inclined  plane  3  feet  high 
and  5  feet  long.     Find  the  components  of  its  weight  along  and  perpendicular 
to  the  plane. 

16.  Explain  how  a  boat  can  sail  almost  in  the  eye  of  the  wind,  by  setting 
the  sail  between  the  direction  of  the  wind  and  the  boat's  course. 

(The  velocity  of  the  wind  may  be  resolved  into  a  component  parallel  to 
the  sail,  which  has  no  effect,  and  a  component  perpendicular  to  the  sail, 
which  exerts  a  pressure  on  it.  This  pressure  may  again  be  resolved  into 
components  parallel  and  perpendicular  to  the  boat's  length.  The  former  is 
the  propelling  force ;  the  latter  causes  leeway,  which  is  made  as  small  as 
possible  by  using  a  keel  or  centreboard  to  resist  sideway  motion.  Draw  a 
diagram  shewing  the  two  resolutions,  and  shew  that  if  P  be  the  pressure 
which  the  wind  would  exert  on  the  sail  if  at  right  angles  to  it,  a,  0  the 
inclinations  of  wind  and  sail  to  the  keel  of  the  boat,  then 

headway  force  =  P  sin  (/3  -  a)  sin  /3, 
leeway  force    =  P  sin  (/3  -  a)  cos  /3.) 

17.  Explain  how  a  kite  is  sustained  in  air,  and  shew  by  a  drawing  that 
the  perpendiculai  to  the  kite  must  lie  between  the  direction  of  the  string 
and  the  vertical. 

18.  Forces  24,  7,  and  25  Ibs.  weight  balance  at  a  point.     Shew  that  two 
of  them  are  at  right  angles. 

19.  A  50  Ib.  weight  is  hung  from  two  points  by  strings  inclined  30°  and 
45°  to  the  vertical.     Find  the  tensions  of  the  strings. 

20.  A  picture  weighing  8  Ibs.  is  hung  by  a  string  passing  over  a  nail 
and  attached  to  two  rings  in  the  top  of  the  frame.     Find  the  tension  of  the 
string  when  the  two  portions  are  inclined  at  an  angle  of  (1)  60°,  (2)  120°. 


XVIT]          THE   COMPOSITION   AND   RESOLUTION   OF   FORCES  171 

21.  Find  the  ratio  of  the  "  power  "  to  the  "  weight "  in  the  inclined  plane 
by  resolving  along  and  at  right  angles  to  the  plane.     Find  also  the  pressure 
on  the  plane. 

(This  is  the  best  way  of  treating  the  inclined  plane,  and  serves  equally 
well  when  the  force  is  not  parallel  to  the  plane.) 

22.  A  weight   W  rests  on  an  inclined  plane,  inclination  a.     Find  the 
force  required  to  sustain  it,  and  the  pressure  on  the  plane, 

(1)  when  the  force  acts  horizontally ; 

(2)  when  its  direction  makes  an  angle  e  with  the  plane  and  above  it. 

23.  Shew  that  if  the  strings  supporting  a  single  moveable  pulley  are 
inclined  at  0  to  the  vertical,  P=  W/2  cos  6. 

24.  Weights  P  and   Q  rest  on  the  upper  edge  of  a  smooth  vertical 
circle,  and  are  connected  by  a  string,  running  round  the  edge,  whose  length 
is  a  quadrant  of  the  circle.     Find  the  position  of  equilibrium,  and  the  tension 
of  the  string. 

(Write  down  the  conditions  of  equilibrium  first  for  P  and  then  for  $,  by 
resolving  along  the  tangent  and  at  right  angles  to  it  in  each  case,  assuming 
that  the  radius  to  P  makes  an  angle  6  with  the  horizontal.  The  resulting 
equations  determine  6  and  T.) 

25.  Two  men  raise  a  cask  weighing  300  Ibs.  from  a  cellar  to  the  street  by 
drawing  it  up  planks  inclined  30°  to  the  horizon  by  means  of  two  ropes 
fastened  to  the  wheels  of  a  dray  in  the  street,  passed  down  the  planks,  under 
and  round  the  barrel,  and  pulled  parallel  to  the  planks.     What  is  the  least 
force  each  man  must  exert? 

26.  A  conical  pendulum  consists  of  a  ball  weighing  5  Ibs.  suspended  by  a 
string  4  feet  long.     If  the  ball  is  projected  so  as  to  describe  a  horizontal  circle 
twice  in  three  seconds,  what  will  be  the  inclination  of  the  string  to  the  vertical, 
and  what  will  be  its  tension  ? 

In  what  time  must  the  ball  revolve  in  order  that  the  string  may  be 
inclined  30°  to  the  vertical? 

(The  acceleration  of  the  ball  to  the  centre  may  be  calculated  by  §  77. 
The  resultant  force  on  the  ball,  in  poundals,  in  order  that  it  may  go  on 
describing  the  circle  is  the  product  of  this  acceleration  into  the  mass  of  the 
ball.  We  may  then  either : 

(1)  express  the  condition  that  this  force  is  the  horizontal  component 
of  the  tension  of  the  string,  while  its  vertical  component  is  equal  to  the 
weight ; 

or  (2)  observing  that  if  this  force  were  reversed  in  direction,  it  would 
be  in  equilibrium  with  the  tension  and  the  weight,  treat  the  problem  as  if 
the  ball  were  at  rest  under  the  weight,  the  tension,  and  the  reversed  resultant 
force.) 


172  MECHANICS  [CHAP,  xvn 

27.  A  conical  pendulum  with  a  string  of  length  I  makes  n  revolutions  per 
second.     Shew  that  the  inclination  of  the  string  is  a,  where  cosa  =      ^       . 

28.  Prove  that  in  the   conical  pendulum  the  time    of   revolution    is 
2?r  ./-,  where  h  is  the  vertical  depth  of  the  revolving  ball  below  the  point 
of  support. 

29.  Apply  this  result  to  the  governor  of  a  steam  engine,  and  shew  that 
for  an  engine  making  60  revolutions  per  minute  the  depth  of  the  balls  below 
the  point  of  support  must  be  about  978  inches. 

30.  Why  is  the  outside  rail  of  a  railway  track  raised  above  the  inside  rail 
at  a  curve  ? 

Shew  that  if  a  train  runs  round  a  curve  of  radius  r  feet  with  velocity  #, 
the  floor  of  the  carriage  should  be  inclined  at  an  angle  whose  tangent  is  v2/gr 
in  order  that  there  may  be  no  lateral  thrust  on  the  rails. 

31.  Shew  that  on  a  5-foot  track,  round  a  curve  of  one-eighth  of  a  mile 
radius,  for  a  mean  velocity  of  30  miles  an  hour,  the  outside  rail  ought  to  be 
raised  between  5  and  6  inches  above  the  level  of  the  inner  rail. 


CHAPTER  XVIII. 

FORCES  ACTING  ANYWHERE  IN  A  PLANE. 
170.    I.     RESULTANT  of  Two  Forces  acting  at  different  points. 


Fig.  78. 

Let  the  forces  P  and  Q  act  on  a  body  at  A  and  B. 

To  fix  the  ideas,  suppose  that  the  body  is  a  flat  board  lying  on 
a  horizontal  table,  and  that  the  forces  are  applied  by  cords, 
attached  to  pins  at  A  and  B,  and  carrying  weights  at  their  free 
ends,  which  hang  over  the  edge  of  the  table. 

Let  the  directions  of  P  and  Q,  produced  if  necessary,  meet  in 
0.  The  effect  of  P  and  Q  will  not  be  altered  if  they  are  applied 
at  0  in  the  same  directions,  instead  of  at  A  and  B. 

This  principle,  that  the  effect  of  a  force  is  the  same,  at 
whatever  point  in  its  line  of  action  it  is  applied,  is  known  as 
the  principle  of  the  Transmissibility  of  Force.  It  may  be  regarded 
as  an  axiom  directly  based  on  experience.  In  fact  we  feel  that  if 
the  cords  by  which  P  and  Q  are  applied  are  prolonged  to  0  and 


174  MECHANICS  [CHAP. 

fastened  to  a  pin  there,  the  pins  at  A  and  B  may  be  taken  out 
without  disturbance. 

(Or  the  principle  may  be  deduced  from  some  other  axiom  of  experience, 
such  as  the  Third  Law  of  Motion.  All  actual  bodies  undergo  slight  changes 
of  shape  on  the  application  of  force.  The  idea  of  a  perfectly  rigid  body  is  a 
mathematical  fiction,  useful  because  most  of  the  solids  known  to  us  approxi- 
mate so  closely  to  it  that  in  Statics,  where  we  are  concerned  with  the  external 
relations  between  different  bodies,  we  can  greatly  simplify  our  theorems  if  we 
are  content  to  ignore  the  very  small  internal  displacements  and  the  corre- 
sponding (often  great)  internal  forces  that  are  called  into  play.  When  these 
are  taken  account  of,  we  enter  on  the  Theory  of  Elasticity. 

Consider,  as  the  simplest  case,  a  fine  rubber  thread  kept  stretched 
by  two  forces  applied  at  the  ends.  Every  particle  of  the  thread  is  drawn 
apart  from  those  on  each  side  of  it  till  the  forces  of  cohesion  so  developed 
are  sufficient  to  prevent  further  stretching.  By  the  Third  Law  the  forces 
between  each  pair  of  particles  are  then  equal  and  opposite.  The  stretching 
force  applied  to  one  of  the  end  particles  must,  for  equilibrium,  exactly 
balance  the  internal  pull  of  the.  second  particle  upon  the  first ;  and  so  on 
throughout  the  string,  till  the  internal  pull  upon  the  last  particle  balances 
the  external  force  applied  to  the  other  end.  The  pull  is  thus  transmitted 
by  the  stretched  thread  so  as  to  balance  the  exactly  equal  pull  at  the  other 
end. 

In  the  case  of  the  elastic  thread  the  displacements  would  be  so  large  that 
they  must  be  taken  account  of ;  but  the  internal  forces,  though  everywhere 
equal  to  the  external  pulls,  may  be  ignored,  since  they  occur  in  equal  and 
opposite  pairs.  The  same  process  goes  on  in  the  solids  contemplated  in 
Statics  as  rigid  bodies.  But  the  displacements  are  so  small  as  not  to  affect 
the  configuration,  and  so  may  be  left  out  of  account.  The  internal  forces 
may  be  ignored  for  the  same  reason  as  before  ;  and  this  is  true  even  when  the 
line  of  action  of  the  transmitted  force  passes  outside  the  body. "  For  let  two 
equal  forces  act  at  A  and  0  (Fig.  79)  in  opposite  directions.  Then  the 
internal  reactions  between  the  parts  of  the  body,  which  hold  it  together,  arc, 
by  the  Third  Law,  at  every  point,  whether  in  the  line  AO  or  elsewhere,  equal 
and  opposite.  Therefore  the  whole  set,  including  the  two  equal  forces  at 
A  and  0,  will  balance,  just  as  they  would  if  both  the  forces  were  applied  at 
the  point  0.  Hence  the  effect  of  a  force  at  A  is  the  same  as  if  it  were  applied 
at  0,  another  point  in  its  line  of  action.) 

The  resultant  of  P  and  Q  acting  at  0  passes  through  0.  Let 
it  cut  the  line  joining  AB  in  C,  and  take  OC,  which  already 
represents  it  in  direction  and  point  of  application,  to  represent  it 
also  in  magnitude. 

Draw    CD,   CE  parallels    to    OB,    OA ;    then    OD,   OE  will 


XYIIl]  FORCES  ACTING  ANYWHERE   IN   A   PLANE  175 

represent  the  components  P  and  Q  on  the  scale  of  the  resultant 

oc. 


Fig.  79. 


Drop  CF,  CG  perpendiculars  to  OA,  OB.     Then,  area  of  the 
parallelogram 

CDOE  =  ODxCF=OEx  CG. 


.-.  PxCF=QxCG. 

Thus  the  resultant  cuts  AB  in  a  point  C  such  that  the  moments 
of  P  and  Q  about  C  are  equal  and  opposite. 

171.  Since  we  might  have  taken  any  length  OC  to  represent 
the   resultant,  this   property   must   hold   for   all   points   on   the 
resultant  ;  or,  what  comes  to  the  same  thing  : 

The  algebraical  sum  of  the  moments  of  two  concurrent  forces 
about  any  point  on  the  line  of  action  of  their  resultant  is  zero. 

172.  The    property  just   proved   is   a   particular   case   of  a 

theorem  communicated  to  the  Paris  Academy  by 

Vangnon  s  _  J       J 

Theorem  of        Varignon  in  1687  (the  year  of  the  Principia  and 

Moments. 

of  Lami  s  theorem). 

The  moment  of  the  Resultant  of  two  co-planar  Forces  about  any 
point  in  their  plane  is  equal  to  the  (algebraical)  sum  of  the  moments 
of  the  Forces. 

Consider  two  forces  AB,  AC  and  their  resultant  AD. 

The  moment  of  the  force  AB  about  the  point  0  is  the  product 
of  AB  by  the  perpendicular  from  0  on  AB,  i.e.  twice  the  area  of 
the  triangle  OAB.  Similarly  the  moments  of  AC,  AD  will  be 
twice  the  triangles  OAC,  OAD. 

We  have  to  shew  that 

OAD  =  OAB  +  OAC 


176  MECHANICS  [CHAP. 

when  0  is  outside  the  angle  between  the  forces  (Fig.  80  a), 
and  OAD  =  OAB -OAG 

when  0  is  inside  (Fig.  80  b). 


Fig.  80. 

The  perpendiculars  from  the  vertices  B,  C,  D  of  the  three 
triangles  OAB,  OAG,  OAD  upon  their  common  base  OA  are 
equal  to  the  projections  of  AB,  AC,  AD  on  a  line  at  right  angles 
toOA. 

But  the  projection  of  AD  on  any  line  is  equal  to  the  algebraic 
sum  of  the  projections  of  AB  and  BD ;  or  of  AB  and  AC  which  is 
equal  and  parallel  to  BD. 

.'.  the  area  OAD  is  equal  to  the  algebraic  sum  of  the  areas 
OAB,  OAG. 

The  4-  sign  is  obviously  to  be  taken  in  Fig.  80  (a),  and  the 
-  in  Fig.  80  (b). 

If  0  is  on  the  line  of  the  resultant,  the  moment  of  the 
resultant  and  therefore  the  sum  of  the  moments  of  the  forces,  is 
zero. 

Note.  The  Moment  of  a  force  about  a  point  may  be 
conveniently  represented  by  the  area  (or  double  the  area)  of  the 
triangle  formed  by  joining  the  point  to  the  ends  of  the  line 
representing  the  force. 


173.  If  the  forces  P  and  Q  (§  170)  are  parallel,  their 
directions  will  not  meet,  and  our  construction 
fails. 


Parallel  Forces. 


XVIII] 


FORCES   ACTING   ANYWHERE   IN   A    PLANE 


177 


This  is  a  case  for  employing  the  principle  of  Continuity, 

us    start    with    the    figure    of    §  170,    and  Q 

gradually   bring    the    forces    to    parallelism,  $ 

making   them   both    approach    the    direction  /| 

perpendicular    to    AB.      The    parallelogram  jj\ 

CEOD    becomes    more    and    more    lozenge-  ;  i\ 
shaped  (Fig.  81),  and  the  diagonal   is  more 
and  more  nearly  equal  to  the  sum  of  the  sides. 
The  law  of  moments, 

PxCF=QxCG, 

remains  true,  but  GF  and  CG  approach  the 
values  CA,  GB. 

We  can  see  what  will  happen  in  the  limit, 
when  the  forces  become  really  parallel. 

(1)  The  Resultant  becomes  the  sum  of 
the  forces,  so  that 


(2)  It  is  parallel  to  the  forces. 

(3)  It  cuts  AB  in  C,  so  that 

PxAG=QxBG. 

It  is  thus  completely  determined. 

When  the  forces  are  parallel,  the  segments 


Let 


p  Q 

Fig.  81. 

of  any  oblique  line  A'CB'  through  G  may  be  used  instead  of 
ACS,  since 

A'C     AC  _Q 

B'G      BC~P' 


.-.  PxAC=QxB'G. 
174.     One  of  the  most  familiar  cases 

The  Principle  of      is  when   the    tw°   forces   OCt 
the  Lever.  on     ft     ^ar     Qr     other     body 

which  is  only  free  to  turn  on  a  pivot.     To 
find  the  relation  between  the  forces  and 
their  distances  so  that  they  should  have 
c. 


A' 


Fig.  82. 


12 


178 


MECHANICS 


[CHAP. 


equal  power  to  turn  the  bar  about  the  pivot  was  Archimedes' 
famous  problem  of  the  Lever.  (§  3.) 

The  difficulty  of  judging  between  unequal  forces  at  unequal 
distances  vanishes  when  only  one  force  is  applied  to  the  bar,  for  it 
will  certainly  turn  it  one  way  or  the  other  unless  the  force  goes 
through  the  pivot. 

But  we  can  now  replace  the  two  forces  by  their  single  resul- 
tant. Then  if  G  is  the  pivot  the  resultant  must,  in  the  case  of 
equilibrium,  go  through  C,  and  therefore,  whether  the  forces 
intersect  or  are  parallel,  if  they  are  to  have  equal  torques,  i.e. 
tendencies  to  turn  the  bar  about  the  pivot,  the  product  of  each 
into  the  perpendicular  distance  from  the  pivot  must  be  the  same. 
This  product  must  therefore  be  the  proper  measure  of  the  torque 
of  a  force  about  a  point.  As  we  have  seen,  Leonardo  called  it  the 
Moment  of  the  force  about  the  point. 

The  principle  of  the  Lever  thus  follows  from  the  Parallelogram 
of  Forces.  We  might  go  on  to  deduce  from  it  all  that  has  been 
given  in  §§  6 — 33  about  the  Centre  of  Gravity,  the  Balance,  Wheel 
and  Axle,  and  the  Pulleys. 


Parallel  Forces 
in  Opposite 
Directions. 


175.     The  student  may  deduce  from  Fig.  83  what  will  happen 
when  the  forces 
act  in  opposite 
directions.  Let 
Q  be  the  larger  ;  then 

(1)  R  =  Q-P. 

(2)  R  is  parallel  to  P  and 
Q  and  cuts  AB  outside,  beyond 
the  larger  force. 

(3)  Since  C  is  on  the  re- 
sultant, by  Varignon's  Theorem, 
the    moments    of    the    forces 
about  C  are  equal  and  opposite. 
If  they  are  perpendicular  to  A  B, 

PxAC=QxBC, 
and  this  is  extended  as  before  to  an  oblique  line  A'CB'. 


83- 


XVIII]  FORCES   ACTING    ANYWHERE    IN    A   PLANE  179 

(Or  the  case  of  unlike  forces  may  be  deduced  from  that  of  like 
forces  as  in  §  13.) 

176.  An  important  case  of  failure  of  the  method  for  finding 

the  resultant  of  two  forces  remains  to  be 
considered.  If  the  forces  are  unlike  in  direction 
and  equal,  the  resultant  Q  —  P  vanishes.  Moreover  no  point 
C  can  be  found,  outside  AB,  which  will  make  P  x  A  G  =  P  x  BC. 
When  Q  is  very  nearly  equal  to  P,  C  has  to  be  a  long  way  off. 
For  equality  G  would  have  to  be  at  an  infinite  distance. 

In  fact  a  pair  of  equal  parallel  forces  acting  in  opposite 
directions  has  no  single  resultant,  and  cannot  be  balanced  by  any 
single  force.  They  have  no  tendency  to  move  a  body  from  one 
place  to  another,  which  could  be  met  by  a  single  force ;  but  they 
tend  to  turn  it  round  in  its  place ;  to  give  it  a  twist.  Such  are 
the  forces  applied  by  the  thumb  and  finger  to  the  wings  of  a  screw 
nut;  or  by  the  hands  to  the  bar  of  a  copying-press;  or  to  a  capstan 
by  two  men  working  on  opposite  sides  of  it. 

A  pair  of  equal,  unlike,  parallel  forces  is  called  a  Couple.  The 
perpendicular  distance  between  the  forces  is  called  the  Arm  of 
the  Couple. 

A  Couple  has  no  single  resultant,  and  no  single  force  can 
balance  it.  But  it  has  a  twisting  tendency,  or  Torque,  measured 
by  its  moment. 

177.  The  theory  of  Couples  was  introduced  by  Poinsot,  and 
The  Properties         affords  a  beautiful  method  of  simplifying  compli- 
Of  couples.  cated  systems  of  forces. 

(In  thinking  about  couples  it  should  be  borne  in  mind  that 
they  are  here  supposed  to  be  applied  to  a  rigid  body  kept  at 
rest  by  certain  forces,  and  that  the  couples  considered  form  part 
of  the  system  of  forces  maintaining  the  equilibrium.  It  is  shewn 
in  works  on  Rigid  Dynamics  that  the  effect  of  a  couple  applied  to 
a  rigid  body  otherwise  free  to  move  is  to  set  it  rotating  about  an 
axis  passing  through  its  centre  of  gravity,  but  not  necessarily 
perpendicular  to  the  plane  of  the  couple.) 

Let  the  forces  in  Fig.  84  be  not  quite  equal;  suppose  the 
force  at  A  to  be  the  weight  of  one  pound  (=  7000  grains),  and 

12—2 


180  MECHANICS  [CHAP. 

that  at  B  to  be  one  pound  and  one  grain.  Let  AB  be  one  foot. 
Then  their  resultant  will  be  a  force  of  one  grain,  directed  upwards 
and  applied  at  a  point  0,  7000  feet  away  to  the  right  of  B. 

If  these  forces  are  applied  to  a  rod  pivoted  at  0,  there  is 
equilibrium.  But  if  the  pivot  be  anywhere  else,  there  will  be 
a  tendency  to  turn  measured  by  the  moment  of  the  resultant 
about  the  pivot.  For  instance,  if  we  take  for  unit  of  moment  the 
moment  of  a  force  of  one  pound  about  a  point  distant  one  foot 
from  its  line  of  action,  and  if  the  pivot  is  20  feet  to  the  right  of  B, 
the  moment  will  be  6980  x  ^m  =  f§f §•  units.  The  moment  about 
a  pivot  10  feet  to  the  right  of  B  is  $fjfj;  about  B  it  is  fggg  =  1 ; 
about  A,  ftlgj. 

It  is  clear  that  for  pivots  anywhere  in  the  neighbourhood  of 
the  forces  the  moments  are  all  very  nearly  equal — equal,  indeed, 
to  the  product  of  the  force  at  B  (one  pound)  by  the  distance  AB 
(one  foot).  To  make  a  difference  pf  so  much  as  one  per  cent.,  the 
pivot  must  be  at  least  70  feet  away  from  B.  In  spite  of  the 
smallness  of  the  resultant  its  moment  remains  considerable  owing 
to  the  distance  at  which  it  acts.  But  then  a  considerable  change 
in  that  distance  is  required  to  produce  any  marked  alteration  in 
the  value  of  the  moment. 

Proceeding  to  the  limit  when  the  forces  are  exactly  equal  (say 
one  pound  each),  we  see  that  (1)  the  resultant  utterly  vanishes  ; 
(2)  the  moment  remains  finite  (equal  to  one  unit);  (3)  the 
moment  is  the  same  wherever  the  pivot  is  placed  in  the  plane  of 
the  couple. 

This  might  have  been  deduced  directly  from  a  consideration  of 
the  forces. 


Fig.  84. 


XVIII]  FORCES   ACTING   ANYWHERE   IN   A   PLANE  181 

For  draw  any  line  AB  cutting  the  forces  at  right  angles,  and 
take  moments  about  a  point  G  in  AB.  Then  if  C  is  between 
A  and  5, 

Moment  of  Couple  =  P  x  AC  +  P  x  BC  =  P  x  AB-y 
and  if  C  is  outside,  as  at  C', 

Moment  of  Couple  =  P  xAC'-Px  EG'  =  PxAB. 

Thus  P  x  AB  is  the  moment  about  any  point  in  AB ;  and  AB 
may  be  drawn  anywhere. 

Hence,  (1)  the  Moment  of  the  Couple  is  the  same  about  every 
point  in  its  plane;  and  is  measured  by  the  product  of  either  of  the 
forces  into  the  perpendicular  distance  between  them,  i.e.  into  the 
arm. 


\r 


P 

•O'  «° 

Fig.  85. 

Consider  two  couples  with  equal  forces  and  arms,  but  in 
different  positions  with  regard  to  a  point  0.  Neither  of  them  has 
any  resultant.  The  only  effect  of  each  is  a  torque,  or  tendency  to 
turn  the  body  about  0;  and  this  is  measured  by  the  product 
of  the  force  and  arm,  which  is  the  same  for  each.  The  couples 
are  therefore  equivalent. 

Or  look  at  it  in  this  way.  It  is  easy  to  find  a  point  0'  which 
is  placed  with  regard  to  the  couple  C  precisely  as  0  is  with 
regard  to  G'.  The  effect  of  C'  about  0  is  then  the  same  as  that 
of  G  about  0';  or  by  (1),  of  C  about  0. 

Hence,  (2)  a  couple  may  be  turned  through  any  angle  without 
altering  its  effect,  and 


182 


MECHANICS 


[CHAP. 


(3)  a  couple  may  be  removed  to  any  other  position  in  the  plane 
without  altering  its  effect. 

(4)  Since  the  whole  effect   of  a   couple  is  measured   by  its 
moment,  a  couple   may  be   replaced   by  any  other   couple   in   the 
same  plane  having  an  equal  moment. 

(Or,  directly  from  the  forces,  supposed  parallel. 


Fig.  86. 
Let  PP,  QQ  be  two  couples  of  equal  moment,  so  that 


but  acting  in  opposite  senses. 

The  resultant  of  P  at  B  and  Q  at  G  is  P  +  Q  acting  at  a  point 

0  such  that 

PxBO=QxOC. 

But  PxAB^QxCD, 


or  PxAO=QxOR 

/.  the  resultant  of  P  at  A  and  Q  at  D  is  P  +  Q  acting  at  the 
same  point  0,  but  in  the  opposite  direction. 

The  couples  therefore  balance,  so  that  a  couple  P  x  AB  is 
equivalent  to  another  couple  Q  x  CD  of  equal  moment  acting  in 
the  same  sense.) 

(5)  Any  number  of  couples  in  a  plane  may  be  replaced  by  a 
single  couple  of  the  same  total  moment. 

Since  the  only  effect  of  a  couple  is  its  torque,  measured  by  its 
moment,  this  follows  from  the  physical  independence  of  forces 


XVlll]  FORCES  ACTING  ANYWHERE   IN  A   PLANE  183 

implied  in  the  Second  Law  of  Motion.  But  for  the  sake  of  the 
importance  of  the  subject  we  will  now  deduce  this  (and  incident- 
ally all  the  previous  propositions)  directly  from  the  parallelogram 
of  forces.  The  student  may  draw  the  figure  for  himself. 


178.  Consider  two  couples  with  forces  PP'  and  QQ',  and  in 
order  to  take  the  most  general  case  let  the  forces  of  the  one  couple 
be  not  parallel  to  the  forces  of  the  other  couple. 

If  the  lines  of  action  of  the  four  forces  be  produced,  they  will 
form  a  parallelogram  A  BCD,  the  sides  AB  and  CD  being  the 
lines  of  action  of  the  forces  P  and  P'  of  the  one  couple,  and  the 
sides  AD  and  CB  being  the  lines  of  action  of  the  forces  Q  and  Q' 
of  the  other  couple. 

But  the  resultant  of  P  and  Q  is  a  force  R  acting  through  A, 
and  having  a  moment  about  any  point  which  is  the  algebraical 
sum  of  the  moments  of  P  and  Q. 

Also  the  resultant  of  P'  and  Q'  is  a  force  R'  acting  at  C,  and 
equal  and  opposite  to  R. 

Thus  the  two  forces  R  and  R'  form  a  couple  which  is  in  all 
cases  the  resultant  of  the  two  original  couples.  And  since  the 
moment  of  a  couple  is  the  sum  of  the  moments  of  its  component 
forces,  we  see  at  once  from  Varignon's  Theorem  that  the  resultant 
couple  has  a  moment  which  is  the  algebraical  sum  of  the  moments 
of  the  original  couples. 

Again,  when  the  moments  of  the  original  couples  are  equal 
and  opposite,  the  resultant  couple  has  no  moment,  that  is,  the 
original  couples  balance  each  other.  In  this  case  R  and  R'  are 
not  only  equal  and  opposite,  but  they  act  along  the  same  line. 

179.  II.     Three  or  more  Forces. 

Any  number  of  forces  acting  at  different  points  in  a  plane  can 
always  be  reduced  to  a  single  force  acting  at  any  point  and  a 
couple. 

Let  P  be  one  of  the  forces,  acting  at  a  point  A.  Take  any 
convenient  origin  0,  and  apply  at  it  two  equal  opposite  forces 
each  equal  and  parallel  to  P.  This  balancing  pair  of  forces  will 
not  alter  the  effect. 


184 


MECHANICS 


[CHAP. 


But  the  three  forces  are  now  equivalent  to 

(1)  a  force  P  acting  at  0,  and 

(2)  a  couple  of  moment  P.p. 


Fig.  87. 

Let  the  same  be  done  for  all  the  other  forces.  The  system 
reduces  to 

(1)  the    forces    transferred   from    their    actual    points    of 
application,  and  all  acting  at  0; 

(2)  a  set  of  couples. 

The  resultant  of  (1)  may  be  found  by  §  164.  The  couples  (2) 
are  equivalent  to  a  single  couple  of  moment  equal  to  the  sum  of 
their  moments. 

We  may  carry  the  simplification  one  step  further,  and  reduce 
the  system  to  a  single  force,  or  a  couple. 

For  if  the  resultant  force  does  not  vanish,  let  the  couple  be 
replaced  by  another  couple  of  equal  moment,  but  with  forces  each 
equal  to  the  resultant  force.  Let  this  couple  be  turned  round  its 
axis  till  one  of  its  forces  "is  directly  opposed  to  the  resultant 
force.  Then  these  two  vanish,  and  the  system  reduces  to  the  other 
force  of  the  couple ;  i.e.  to  a  single  force. 

In  the  case  where  the  resultant  force  vanishes  the  system 
obviously  reduces  to  the  couple. 


XVIII] 


FORCES   ACTING   ANYWHERE   IN   A   PLANE 


185 


180.  It  is  often  more  convenient  to  choose  two  axes  of 
reference,  and  to  resolve  each  force  into  its  components  in  these 
directions  before  transferring  to  the  origin.  Thus : 


X 


y 

Fig.  88 

Let  the  position  of  p,  the  point  of  application  of  the  force  P, 
fixed  by  the  co-ordinates  ON=x,  Np  =  yt  and  let  the  com- 
ponents of  P  parallel  to  the  axes  be  X  and  Y. 

Then  X  may  be  replaced  by 

(1)  X  acting  at  0  along  Ox,  and 

(2)  a  couple  —Xy. 
Y  may  be  replaced  by 

(1)  Y  acting  at  0  along  Oy,  and 

(2)  a  couple  +  Ycc. 

Let  the  same  be  done  for  all  the  forces,  and  let  us  indicate  the 
sum  of  all  products  like  Xy  by  ^Xy. 

The  system  reduces  to  2JT  and  2F  acting  at  0;  and  a  set  of 
couples  2  (  Yx  —  Xy). 

The  resultant  of  %X  and  SF  will  be  a  force  R,  and  a  single 
couple  G  may  be  found  whose  moment  is  equal  to  S  (  Yx  —  Xy). 


186  MECHANICS  [CHAP. 

181.  I.     Two  Forces. 

These    must   act   along  the   same    line    in 

Conditions  of  Equili- 

brium for  any  Forces       opposite    directions,    and    be     equal    in    mag- 

in  one  Plane.  *  1     _ 

nitude. 

182.  II.     Three  Forces. 

If  three  forces  are  in  equilibrium,  they  either  meet  in  a  point, 
or  are  parallel. 

For  if  two  of  them  meet,  their  resultant  goes  through  their 
meeting  point,  and  can  only  be  balanced  by  a  force  passing 
through  that  point;  but  if  two  of  them  are  parallel,  their  resultant 
is  parallel  to  them,  and  therefore  so  is  the  force  which  is  to 
balance  them. 

Very  many  problems  occur  in  which  the  lines  of  action  of  two 
out  of  the  three  forces  are  known,  and  the  solution  depends  on 
making  the  third  force  pass  through  their  point  of  intersection. 
The  magnitudes  of  the  forces  are  then  generally  given  by  Lami's 
Theorem,  or  some  geometrical  application  of  the  Triangle  of 
Forces.  In  the  case  of  parallel  forces  we  have 

R  =  P  +  Q 
and  PxAC=Qx£0. 

183.  III.     Any  number  of  Forces. 

Let  them  be  reduced  to  a  single  force  R  acting  at  any  point  ; 
and  a  couple  G. 

Since  a  Couple  cannot  be  balanced  by  a  force,  it  is  necessary 
for  equilibrium  that  both  the  force  and  the  couple  should  vanish 
separately. 

Thus 

R  =  0\ 

0-OJ* 

These  are  equivalent  to 


and 

i.e.  (1)  the  sums  of  the  resolved  parts  of  all  the  forces  in  any  two 

directions  at  right  angles  must  be  separately  zero;  and 


XVIIl]  FORCES   ACTING   ANYWHERE   IN    A   PLANE  187 

(2)  the  sum  of  the  moments  of  all  the  forces  (or  of  their  com- 
ponents} about  any  point  must  vanish. 

The  conditions  (1)  secure  that  there  shall  be  no  movement 
of  translation,  while  (2)  must  be  satisfied  if  there  is  to  be  no 
rotation. 

184.     The  general  procedure  in  solving  a  Statical  problem 

Method  of  Solving       iSHSfolloWS: 

statical  Problems.  (i)  Draw  a  figure  and  see  that  all  the  external 

forces,  as  well  as  any  internal  reactions  that  are  to  be  considered, 
are  properly  represented. 

(2)  Select  some  body  or  system  of  bodies,  and  write  down  the 
conditions  of  equilibrium  for  it;  i.e.  choose  any  convenient  axes 
at  right  angles ;  resolve  the  forces  acting  on  the  system  along 
them,  and  equate  the  sums  of  the  resolved  parts  to  zero;  and 
"  take  moments"  about  any  convenient  point,  equating  the  sum 
of  the  moments  to  zero. 

It  is  most  important,  as  in  Dynamics  (§  128),  to  settle  quite 
definitely  what  is  to  be  the  system  considered  when  writing  down 
each  equation. 

The  two  equations  of  the  resolved  parts,  and  the  one  of 
moments,  in  general  suffice  to  determine  two  co-ordinates  of  some 
point  in  each  body,  and  one  angle  fixing  its  azimuth  about  the 
point.  For  each  unknown  reaction,  such  as  a  pressure  between 
bodies  in  contact,  or  the  tension  of  a  string,  a  geometrical  relation 
can  be  written  down;  so  that  there  will  be  as  many  equations  as 
quantities  to  be  found. 

By  choosing  axes  at  right  angles  to  some  of  the  forces,  and  by 
taking  moments  about  points  through  which  their  directions  pass, 
we  can  often  prevent  forces  whose  values  are  unknown,  or  not 
wanted,  from  entering  into  the  equations,  and  obtain  what  we 
want  from  one  of  the  equations  of  resolution,  or  from  the  equation 
of  moments  alone.  A  proper  choice  of  axes  and  points  for  taking 
moments  will  greatly  simplity  most  problems. 


188 


MECHANICS 


[CHAP. 


EXAMPLES. 

1.  A  uniform  beam  of  weight  W  can  turn  about  a  hinge  at  one  end  A, 
and  is  drawn  aside  from  the  vertical  by  a  horizontal  force  P  applied  to  the 
other  end  B.  Find  the  position  of  equilibrium  and  the  reaction  at  the  hinge. 

Consider  the  beam.  Three  forces  act  on  it,  P,  W,  and  the  reaction  R  at 
the  hinge.  Hence  R  must  go  through  the  meeting  point  of  P  and  W. 

Let  2a  be  the  length  of  the  beam  ;  #,  <£  the  inclinations  to  the  vertical  of 
the  beam  and  of  the  reaction  R. 


It  * 

T 
>p 


(1)    By  Lami's  Theorem 


W 


sin  <£     cos 


Whence  72  is  known. 
By  Geometry 

tan  <p 


R 

~  sin  90° " 
P 


CD     AE      asm0      ..       . 

=  --^  =  -— ^  = A  =  4  tan  6. 

AC     AC     2acos<9     " 

2P 
.'.  tan  6  =  2  tan  (ft  =  - . 


XVIII]  FORCES   ACTING    ANYWHERE   IN   A   PLANE  189 

(2)     By  the  general  conditions  of  equilibrium. 

Without  necessarily  assuming  that  R  goes  through  D,  we  have,  by  resolving 
horizontally  and  vertically, 

R  sin  <£  =  P, 

R  cos  0  =  IF, 
whence  R  and  $  are  known. 

The  inclination  of  the  beam  is  found  directly  by  taking  moments  about  A, 
when  R  does  not  come  in.     Thus 

P  .  2a  cos  Q  =  W  .  a  sin  6 

2P 

and  tan  6  =        . 


2.  A  beam  of  length  2«  and  weight  W  turns  on  a  hinge  at  one  end  A,  and 
is  supported  at  an  inclination  0  to  the  vertical  by  a  string,  attached  to  its 
other  end  B.  The  string  passes  over  a  smooth  pulley  at  (7,  and  sustains  a 
weight  P.  C  is  vertically  above  the  hinge  A,  and  distant  b  from  it.  Shew  that 

(4P2  _  JF2)  62  -  4  TF2  a2 


3.  One  end  A  of  a  beam  AB,  weight  TF,  rests  against  a  smooth  vertical 
wall,  and  a  cord  attached  to  the  other  end  B  passes  over  a  smooth  pulley 
at   0,  vertically  above  A,  and  sustains  a  weight   P.      Find  the  pressure 
on  the  wall. 

4.  A  ladder  30  feet  long  and  weighing  56  Ibs.  rests,  at  an  inclination  of 
60°  to  the  horizon,  with  its  lower  end  against  the  foot  of  a  wall.     It  is 
sustained  by  a  rope  attached  to  a  rung  20  feet  from  the  bottom,  and  carried 
horizontally  to  a  staple  in  the  wall.     The  centre  of  gravity  of  the  ladder  is 
12  feet  from  the  bottom.     Find  the  tension  of  the  rope,  and  the  direction 
of  the  reaction  at  the  bottom  of  the  ladder. 

What  would  the  tension  become  if  a  man  weighing  160  Ibs.  climbed  out 
to  the  top  of  the  ladder  ? 

5.  A  beam  is  placed  across  a  smooth  horizontal  rail,  and  rests  with  its 
lower  end  against  a  smooth  vertical  wall,  the  distance  of  which  from  the  rail 
is  one-sixteenth  of  the  length  of  the  rod  ;  shew  that  the  rod  must  be  inclined 
60°  to  the  horizon. 

6.  A  plank  12  feet  long  and  weighing  30  Ibs.  stands  on  the   ground, 
leaning  against  a  smooth   horizontal  rail  at  an  inclination  of  60°  to  the 
ground,  with  3  feet  of  its  length   projecting  beyond  the   rail.     Find  the 
direction  and  magnitude  of  the  reaction  on  the  ground. 


190 


MECHANICS 


[CHAP. 


7.  A  rod  whose  centre  of  gravity  divides  it  into  two  parts  a  and  b  is 
placed  inside  a  smooth  sphere  of  such  a  radius  that  the  rod  subtends  an 
angle  2a  at  the  centre.  Find  the  inclination  6  of  the  rod  to  the  horizon. 

(This  is  a  type  of  many  problems  in  which  beams  or  other  heavy  bodies 
are  supported  by  two  forces.  We  have  done  with  the  mechanics  of  the 
question  the  moment  we  see  that  the  c.G.  must  be  vertically  under  the 
meeting  point  of  the  forces,— in  this  case  the  centre  of  the  sphere. 


/ 


If  we  drop  perpendiculars  from  the  ends  of  the  rod  on  thje  vertical  through 
6r,  the  fact  that  this  vertical  pasMf  through  the-centre  at  once  gives  the 
result  that 

sin  (a  —  6)  :  sin  (a  +  6}  —  a  :  6, 
and  finally 

tan  6  =  , tan  a. 

b  4-  a 

Once  6  is  known,  Lami's  Theorem  gives  the  pressures.) 

8.  A  beam  of  weight  W  whose  centre  of  gravity  divides  it  into  portions 
a  and  6,  rests  with  its  ends  on  two  smooth  planes  inclined  towards  each  other 
and  making  angles  a,  /3  with  the  horizontal.     Find  the  inclination  of  the 
beam  to  the  horizon  and  the  pressures  on  the  planes. 

(Solve  as  in  Question  7,  and  also  solve  by  resolving  the  forces  and  taking 
moments.) 

9.  A  beam  whose  centre  of  gravity  divides  it  into  two  portions  a  and  b 
is  suspended  by  a  rope  of  length  I  attached  to  its  ends  and  passed  over 
a  smooth  peg.     Find  the  inclination  of  the  beam  to  the  vertical  and  the 
tension  of  the  rope. 


XVIIl]  FORCES    ACTING   ANYWHERE   IN   A   PLANE  191 

10.  A  sphere  of  weight   W  rests  between  a  vertical  plane  and  a  plane 
inclined  at  an  angle  a  to  the  horizon.     Find  the  pressures  on  the  planes. 

11.  Two  spheres,  each  of  one  inch  radius,  rest  inside  a  sphere  of  3  inches 
radius.     Shew  that  the  pressure  between  the  small  spheres  is  one-half  the 
pressure  of  either  small  sphere  on  the  large  one. 

12.  A  sphere  is  hung  from  a  hook  in  a  vertical  wall  by  a  string  equal  in 
length  to  the  radius.     Find  the  inclination  of  the  string,  its  tension,  and  the 
pressure  on  the  wall,  if  the  sphere  weighs  10  Ibs. 

13.  A   square  is  hung  up  with  its  plane  perpendicular  to  a  smooth 
vertical  wall  by  a  string  attached  to  one  corner  and  to  a  hook  in  the  wall, 
the  length  of  the  string  being  equal  to  a  side  of  the  square.     Shew  that  the 
distances  of  the  three  corners  from  the  wall  are  as  1  :  3  :  4.  . 

14.  A  carriage  wheel  of  radius  r  and  weight  W  is  to  be  dragged  over  an 
obstacle  of  height  h  by  a  horizontal  force  applied  to  the  centre  of  the  wheel. 
Shew  that  the  force  must  be  slightly  greater  than 

\/2rA  -  A2 


W. 


r  — 


15.  A  step  ladder  weighing  40  Ibs.  has  two  equal  legs,  each  10  feet  long, 
hinged  at  the  top  and  joined  by  a  cord  6  feet  long  at  the  bottom.     It  rests  on 
a  smooth  plane  and  a  weight  of  160  Ibs.  is  placed  on  the  top.     Find  the 
tension  of  the  cord. 

16.  One  end  of  a  uniform  ladder,  84  Ibs.  weight,  rests  against  a  smooth 
vertical  wall  at  a  height  of  12  feet  from  the  ground,  and  the  other  rests  on 
the  ground  10  feet  from  the  wall.     Find  the  pressure  on  the  ground. 

17.  The  two  legs  of  a  light  step  ladder  are  connected  by  a  smooth  joint 
at  the  top  and  a  cord  at  the  bottom.     The  ladder  stands  on  a  smooth  floor 
with  one  leg,  which  is  3  feet  long,  vertical.     A  man  of  11  stone  weight  stands 
on  the  other  leg  at  a  height  of  2  feet  above  the  ground.     Find  the  pressure 
on  the  vertical  leg.     What  is  the  tension  of  the  cord  ? 

18.  The  sides  of  a  triangular  framework  are  13,  20,  and  21  inches;  the 
longest  side  rests  on  a  smooth  horizontal  table,  and  a  weight  of  63  Ibs.  is 
suspended  from   the    opposite    angle.      Find  the  tension  in  the    side    on 
the  table. 

19.  Two  small  heavy  rings  of  weights  W  and  W ',  connected  by  a  light 
string,  slide  on  two  wires  in  the  same  vertical  plane  making  equal  angles 
a  with  the  horizon.     Shew  that  the  inclination  6  of  the  string  to  the  horizon 
is  given  by 


192  MECHANICS  [CHAP,  xvm 

20.  Forces  of  3,  5,  7,  9  Ibs.  weight  act  along  the  sides  AB,  EC,  CD,  DA 
of  a  square,  each  of  the  sides  being  one  foot  long.     Find  their  resultant. 

(Assume  that  it  cuts  the  sides  AB,  AD  at  points  distant  x  and  y  from  A. 
Then  express  the  conditions  that  the  moments  about  these  points  are  zero. 
This  will  determine  two  points  on  the  resultant.  Shew  that  it  is  parallel  to 
the  diagonal  AC.) 

21.  Forces  of  1,  2,  3,  4,  5,  6  Ibs.  weight  act  along  the  sides  AB,  EC,  &c. 
of  a  regular  hexagon,  taken  in  order.     Find  the  single  force  acting  at  A  arid 
the  moment  of  the  couple  which  together  are  equivalent  to  the  system. 

22.  The  resultant  of  three  forces  P,  Q,  R,  which  act  along  the  sides  of 
a  triangle  ABC  taken  in  order,  passes  through  the  centres  of  the  circumscribed 
and  inscribed  circles.     Shew  that 

P         =          Q          =          R 

cos  B  —  cos  C     cos  C  —  cos  A     cos  A  —  cos  B ' 

23.  Four  forces  are  completely  represented  by  the  sides  AB,  AD,  CB, 
CD  of  a  quadrilateral  ABCD.     Shew  that  they  are  equivalent  to  a  couple 
consisting  of  two  forces  through  A  and  C  each  equal  and  parallel  to  the  other 
diagonal. 

24.  Solve  Question  20  by  finding  the  single  force  acting  at  A,  and  the 
couple  which  together  with  it  is  equivalent  to  the  system.     Then  change  the 
couple  to  another  of  the  same  moment  with  forces  each  equal  to  the  single 
force ;  and  turn  it  round  A  till  one  of  its  forces  balances  the  single  force, 
leaving  the  other  as  the  resultant. 


CHAPTER  XIX. 


FRICTION. 

185.  ON  the  principle  of  dealing  with  one  difficulty  at  a  time 
(§  101)  we  have  hitherto  supposed  that  the  surfaces  of  the  bodies 
in  contact  were  perfectly  smooth,  so  that  the  only  possible  reaction 
between  them  was  a  direct  thrust  perpendicular  to  the  surface  of 
contact.     This  is  never  the  case  in  practice.     Wherever  there  is 
tendency  to  motion   of  one   body  which   presses  upon   another, 
friction  is  called  into  play  to  oppose  it. 

186.  The  effects  of  friction  were  first  investigated  by  Coulomb. 
He  used  a  weighted  slider  set  upon  a  horizontal  surface,  with  a 
cord  from  it  carried  over  a  pulley  to  support  a  scale-pan. 


Fig.  89. 

When  no  weight  is  attached  to  the  string,  the  upward  pressure 
from  the  table  balances  the  weight  of  the  slider,  and  the  system 
remains  at  rest.  If  weights  be  gradually  placed  in  the  scale-pan, 
the  slider  at  last  gets  into  motion,  and  then  proceeds  with 
accelerated  velocity. 

Thus  up  to  a  certain  limit  just  enough  friction  is  called  into 
play  to  prevent  motion.  If  the  weight  P  exceeds  this  limit, 
motion  must  ensue,  and  since  it  is  accelerated,  the  friction  during 
c.  13 


194  MECHANICS  [CHAP. 

motion  is  slightly  less  than  it  was  before  starting.  Coulomb 
accordingly  made  his  measure ments  by  first  starting  the  slider, 
and  then  finding  the  weight  P  required  to  keep  it  in  uniform 
motion. 

He  deduced  from  his  experiments  the  following  laws  for  the 
limiting  friction: 

(1)  The  friction   varies   as   the   normal    pressure  when   the 
materials  of  the  surfaces  in  contact  remain  the  same. 

(2)  The  friction  is  independent  of  the  extent  of  the  surfaces 
in  contact  so  long  as  the  normal  pressure  remains  the  same. 

(3)  The  friction  is  independent  of  the  velocity  when  the  body 
is  in  motion. 

These  laws  are  only  approximately  true,  and  should  not  be 
relied  on  for  values  of  the  normal  pressure  or  of  the  velocity 
outside  the  limits  of  those  employed  in  the  experiments  from 
which  the  friction  was  determined.  But  if  we  may  assume  their 
truth,  then  the  friction  is  given  by  the  equation  J?  =  /j,R.  /JL  is 
called  the  coefficient  of  friction  for  the  given  materials. 

The  best  way  to  form  an  idea  of  the  value  of  fj,  is  to  make 
actual  experiments  with  the  simple  apparatus  of  Coulomb. 
Kankine  gives  the  results  of  some  experiments  as  follows : 

For  iron  on  stone  p  varies  between  '3  and  7, 
timber  on  timber  „  „  '2  „  *5, 
timber  on  metals  „  „  "2  „  *6, 
metals  on  metals  „  „  15  „  '25. 

187.     Let  a  body  be  placed  on  a  horizontal  plane,  and  let  the 
plane  be  gradually  tilted  till  the  body  is  just  on 
the  Point  of  sliding.     The  friction  will  act  up  the 
means  of    plane  and  be  /u,  times  the  pressure. 

Resolving  along  and  perpendicular  to  the  plane, 
we  have 

W  sin  a  =  fj,R, 

W  cos  a  =  R, 

.'.     /z,  =  tana. 
The  angle  a  is  called  the  angle  of  friction.    Up  to  the  moment 


XIX] 


FRICTION 


195 


when  the  body  slides,  the  resultant  reaction  of  the  plane  on  the 
body  must  evidently  be  along  the  vertical  to  balance  W.     The 


Fig.  90. 

greatest  angle  the  resultant  reaction  can  make  with  the  normal 
to  the  plane  (when  all  the  friction  possible  is  called  into  play)  is 

a  =  tan"1  p. 

Here,  as  well  as  in  the  method  of  §  186,  it  is  better  to  start  the 
body  sliding  and  find  the  right  slope  of  the  plane  to  keep  the 
velocity  constant.  For  the  friction  between  bodies  that  have 
been  resting  in  contact  for  some  time  varies  irregularly  according 
to  many  other  circumstances. 

188.  The  treatment  of  problems  when  the  friction  is  taken 
into  account  involves  no  new  principles.  But  an  extra  force  must 
be  indicated  in  the  figure,  opposed  to  the  direction  in  which 
motion  is  about  to  commence. 

A  very  convenient  way  of  including  these  forces  is  to  draw 
the  cone  of  friction,  that  is, 


the  cone  of  angle 

2a  =  2tan->, 

about  the  normal  at  the 
point  of  contact  as  axis.  It 
is  clear  that  the  resultant 
reaction  may  lie  anywhere 
inside  this  cone,  but  not  out- 


v" 


Fig.  91. 


13—2 


196 


MECHANICS 


[CHAP. 


side  of  it.     This  consideration  often  serves  to  solve  a  problem  by 
inspection. 

For  example,  a  beam  is  at  rest  but  on  the  point  of  motion, 
inside  a  rough  sphere,  in  a  vertical  plane  through  the  centre. 
To  find  its  inclination  6  to  the  vertical. 


B 


Fig.  92. 

Let  /Lt  =  tana  be  the  coefficient  of  friction  for  the 'beam  and 
the  sphere.  Draw  the  cones  of  friction  at  A  and  B.  Then  since 
all  the  friction  is  called  into  play  to  prevent  motion,  the  resultant 
reactions  at  A  and  B  must  lie  along  the  edges  AC,  BG  of  the 
cones.  The  only  other  force  acting  on  the  beam  is  its  weight.  The 
three  forces  must  meet  in  a  point  (§  182),  so  that  the  vertical 
through  G  passes  through  the  centre  of  gravity.  Let  the  beam 
subtend  an  angle  2/3  at  the  centre  of  the  circle. 
Then 

tACG  =  0-  *CAG  =  0-  (90°  -  0  -  a.)  =  a  +  0  +  6  -  90°. 
Z  BGG  =  180°  -  6  -  Z  CBG  =  180°  -  6  -  (90°  -  0  +  a) 

=  90°  -  (a  -  ft  +  0). 


XIX]  FRICTION  197 

If  the  C.G.  divide  the  beam  in  the  ratio  ra :  n,  we  see,  by  drawing 
horizontals  through  A  and  B  to  the  vertical  CGW,  that 
m  =  AC  sin  ACG  _  AC.  cos  (a  +  ft  +  0) 
n  ~  BC  sin  BCG  ~~  BC .  cos  (a  -  ft  +  0} 

cos  (a  -  ft)  cos  (a  +  0  +  fl) 
"  cos  (a  +  ft)  cos  (a  -  ft  +  0) ' 
whence  0  may  be  found. 

189.  In  the  Simple  Machines  friction  plays  a  very  important 
part  and  cannot  be  neglected.     In  the  case  of  the  Screw  and  the 
Wedge  its  effects  render  the  ordinary  formulae  practically  useless. 
Recourse  must  then  be  had  to  direct  experiment. 

The  student  is  strongly  recommended  to  consult  Sir  Robert 
Ball's  Experimental  Mechanics  for  an  admirable  treatment  of  the 
principal  machines  with  friction,  with  apparatus  and  weights  on  a 
practical  scale.  He  will  learn  much  besides  from  this  book,  as 
well  the  scientific  method  of  dealing  with  sets  of  observations  on 
variable  quantities  like  friction,  as  the  habit  of  keeping  concrete 
facts  in  view  when  studying  principles. 

190.  One  general  consideration  may  be  mentioned.     A  part 
of  the  work  done  by  the  force  applied  to  a  machine  is  absorbed 
in  overcoming  the  friction.     It  is  converted  into  other  forms  of 
energy,  not  destroyed.    But  it  is  lost  for  mechanical  purposes.    The 
remainder  is  the  useful  work  done  by  the  machine. 

Let  EI  =  the  "  lost "  work. 
Eu  =  the  useful  work. 
Pj  =  the  force  required  to  balance  the  weight  when  the 

machine  works  forwards. 
Pt  =  the  force  when  the  machine  works  backwards. 

a  =  the  distance  moved  by  the  "  Power  "  handle. 
Then  P,a  =  Eu  +  Eh 

-P2ot  =  Eu  —  EI, 

for  when   the   machine  works   backwards,  the  friction  aids  the 
"  Power." 
Thus 


198 


MECHANICS 


[CHAP. 


This  fraction  which  expresses  the  ratio  of  the  useful  work 
done  to  the  total  work  done  by  the  "Power"  is  called  the  Efficiency 
of  the  machine. 

If  the  efficiency  =|,  P2  is  zero,  so  that  no  force  is  required 
to  prevent  the  machine  from  running  backwards,  as  friction  is 
sufficient  in  itself  to  stop  it ;  and  this  is  a  fortiori  the  case  for 
smaller  values  of  the  efficiency. 

For  example,  the  Differential  Pulley,  as  usually  constructed, 
has  an  efficiency  less  than  one  half.  The  chain  may  thus  be  let 
go  at  any  stage  without  risk  of  the  machine  running  backwards. 
This  useful  property  compensates  for  the  waste  of  more  than  half 
the  work  done. 

It  is  on  this  property  that  the  usefulness  of  the  Wedge 
depends;  for  only  in  virtue  of  friction  can  it  be  driven  in  by  a 
series  of  blows,  since  otherwise  it  would  slip  back  between  each 
blow  and  the  next. 

This  instrument  is  a  double  inclined  plane.  The  "  Power  "  is 
applied  parallel  to  the  common  base,  and  the 

The  Wedge.  r*  ...  .,  i        j  •    •  j    j 

resistance  or  the  material  to  be  divided  acts  as 
a  pressure  on  the  two  faces. 

Let  2a  be  the  angle  of  the  wedge ;  At  the  coefficient  of  friction. 
Then  by  resolving  along 
the  central  line, 

P  =  2R  sin  a  4-  2j*R  cos  a     . 
and 


-5  =  2  (sin  a  +  ft  cos  a). 
JK 

Apart  from  friction  it  is 
clear  that  by  making  a  very 
small  we  may  reduce  this 
ratio  as  much  as  we  please, 


Fig.  93. 


so  that  a  very  small  force  will  overcome  a  great  resistance. 

In  cutting  or  piercing  instruments  the  edges  or  points  are 
Theory  of  the  ground  down  to  an  excessively  small  angle.  The 
Knife-  mechanical  advantage  may  be  still  further  in- 

creased by  pushing  or  drawing  the  knife  along,  instead  of  pressing 
it  in  perpendicularly. 


XIX]  FRICTION  199 

For  if  the  pressure  be  applied  in  the  direction  D'C  (Fig.  94), 
inclined  at  an  angle  /3  to  the  edge  of  the  knife,  instead  of  along 


A 


A' 


Fig.  94. 

DC,  the  effective  angle  of  the  edge  is  A'CB'  instead  of  ACB,  and 
this  may  be  made  as  small  as  we  choose  by  sufficiently  inclining 
D'C. 

If  friction  is  neglected,  the  formula  will  in  this  case  be 

p 

-^  =  2  sin  a. .  sin  /3. 

For  the  knife  is  supposed  to  be  actuated  by  an  oblique  thrust 
P  along  D'C,  and  a  pressure  F  at  right  angles  to  D'C,  applied  by 
smooth  guides  which  compel  it  to  travel  in  this  direction,  or  by 
the  hand  itself.  The  forces  F  and  R  cos  a  are  at  right  angles  to 
CD',  hence  resolving  along  CD'  we  obtain  the  above  equation. 

(Or  by  Lami's  Theorem 

P          F    =  2R  sina 

sin  j8  ~"  cos  |8  ~  sin  90°  ' 
p 

and  therefore  -^  =  2  sin  a .  sin  /3. 
zi 

The  same  result  follows  from  the  principle  of  Work.  For  if  the  knife  be 
moved  obliquely  along  D'C  till  it  has  sunk  in  to  a  vertical  depth  DC,  no  work 
is  done  by  the  pressure  Ft  or  by  the  forces  R  sin  a,  since  their  points  of 
application  remain  at  the  same  vertical  height  throughout.  Hence  the 
work  done  by  P  is  equal  to  that  done  against  the  two  thrusts  R  cos  a.  Now 
these  forces  have  their  points  of  application  separated  by  a  distance  AB}  so 
that  the  work  done  against  them  is  R  cos a.AB. 


200 


MECHANICS 


[CHAP. 


Therefore  P  .  CD'  =  R  cos  a  .  AB, 

an  equation  which  at  once  leads  to  the  formula 


Another  case  of  great  practical  importance  is  that  of  a  rope 
wound  round  a  rough  post. 

191.     Let  a  rope  be  passed  round  a  rough  circular  post  and 

pulled  at  one  end  by  a  force  T,  while  being  held 

wound0nro0unndrpe     back   by  a  force  TQ  at   the  other.     We  will  in- 

posf  h  circular          vestigate  the  relation  between  T  and  T0  in  the 

case  when  TQ  is  barely  sufficient  to  prevent  the 

rope  from  slipping.     Let  the  part  of  the  rope  in  contact  with  the 

post  subtend  an  angle  a  at  the  centre. 

R.PQ 


Fig.  95. 
Consider  a  short  length  of  the  rope  PQ  subtending  a  small 

angle  e^  =  -  at  the  centre. 
ft 

Let  R  be  the  normal  pressure  per  unit  length  of  the  rope. 
Then  the  pressure  on  PQ  is  R  x  PQ,  and  the  friction  is  jj,R .  PQ. 
Let  the  tensions  at  P  and  Q  be  TI  and  Tz. 


XIX]  FRICTION  201 

Resolving  along  the  tangent  and  normal  at  P  we  have 
Tz  .  cos  a^T.  +  ^R.  PQ, 
T,  .  sin  ai  =  R  .  PQ. 

Now  ultimately,  when  Oj  is  very  small,  cos  ^  —  1,  and  sin  fl^  =  c^. 
Also,  if  a  is  the  radius  of  the  post,  PQ  =  ao^. 

Thus  27a  =  T1  +  /*.ZTaa1. 

In  the  term  pT^  we  may  put  Tt  for  T2,  since  the  small 
difference  between  them,  when  multiplied  by  the  small  quantity 
«!,  will  be  negligible  compared  with  the  value  of  T&  itself. 
Hence 

r.-r,  (1 


Let  the  tension   at  the  end   of  the   next  short   length   QS, 
subtending  ax  at  the  centre,  be  Ts.     Then,  as  above, 


Proceeding  in  this  way  we  see  that  if  we  start  from  a  place 
where  the  tension  is  T0  (say,  where  the  rope  first  touches  the  post), 
the  tension  T  at  any  other  point  at  an  angular  distance  a  from  the 
start  is  given  by 


n  J 


Writing  -  for  the  small  fraction  —  ,  we  have 


If  «!  is  taken  very  small,  n  and  x  become  very  large,  and  in 
the  limit,  for  which  alone  the  above  reasoning  is  accurate,  n  and  x 
become  infinitely  great. 

Now  hv  the  Trinomial  Theorem 


3ome  infinitely  great. 
Now  by  the  Binomial  Theorem 

/        IV*  1      x.x  —  1     1 


1      x.x—  \  .x  —  2     1 


1.2  '  1.2.3  +  '" 


202  MECHANICS  [CHAP. 

when  x  is  infinitely  large,  since  the  part  of  each  term  neglected  is 
infinitely  small  compared  with  what  is  retained. 

It  is  proved  in  books  on  Algebra  that  this  series  is  convergent, 
and,  as  more  and  more  terms  are  taken,  constantly  approaches  a 
finite  limit  between  2  and  3  in  value.  In  fact  it  is  the  base  of 
the  natural  system  of  logarithms,  commonly  denoted  by  e,  and 

has  the  value 

6  =  271828.... 

Thus  T=T*<r. 

Hence  the  tension  rapidly  increases  with  a,  and  if  the  rope  is 
taken  two  or  three  times  round  the  post,  an  enormous  force  will 
be  required  to  make  it  slip,  although  TQ>  the  force  applied  to 
prevent  motion,  may  be  very  small.  This  is  the  principle  em- 
ployed in  bringing  boats  to  their  moorings.  A  man  with  a  few 
turns  of  rope  round  a  post  can  hold  up  a  great  steamer,  if  the  rope 
be  strong  enough. 

192.  The  relation,  discussed  above,  between  the  tension  and 
the  angle  is  worth  careful  consideration,  for  the  student  of  Physics 
will  meet  with  many  similar  cases.  The  essential  feature  is  that 
while  the  angle  increases  in  arithmetical  progression  the  tension 
increases  in  geometrical  progression.  This  is  the  relation  between 
a  power  of  a  number  and  the  corresponding  index ;  or  between  a 
number  and  the  corresponding  logarithm;  so  that  it  may  be 
described  as  the  law  of  logarithmic  growth. 


The  formula  T  =  TQ  ( 1  +  ^ 

\         n ; 


is  exactly  analogous  to  that  for  the  amount  of  a  sum  of  money  P0 
invested  for  n  years  at  r  per  cent,  compound  interest, 


So  far,  the  increase  takes  place  in  a  series  of  finite  steps, 
corresponding  to  the  addition  of  definite  small  angles  in  the  one 
case,  and  to  whole  years  in  the  other.  In  commerce  the  interest 
is  sometimes  added  to  the  principal  more  frequently  than  by 
years — say  quarterly.  But  in  physics  we  have  to  go  a  step 
further,  and  suppose  that  the  increment  is  instantly  added  to  the 


XIX]  FRICTION  203 

principal  as  it  accrues.     The  calculation  of  the  last  article  shews 
that  the  formula  then  becomes 


This  law  is  met  with  whenever  the  rate  of  change  of  a  quantity 
is  proportional  to  the  amount  of  it  already  existing. 

In  physics  the  law  is  more  frequently  found  governing  the 
rate  of  decrease  at  which  some  phenomenon  dies  away  ;  as  in 
the  case  of: 

(1)  The  gradually  decreasing  swings  of  pendulums  and  galvano- 
meter needles.     The  loss  due  to  resistances  is  proportional  to  the 
length  of  swing  over  which    they   act,   so    that   each   swing   is 
diminished  by  a  definite  fraction  from  the  last.    Hence  in  accurate 
work  with  the  Ballistic  Pendulum  of  §  269,  it  would  be  necessary 
to  find  what  is  called  the  logarithmic  decrement  by  observing  the 
proportional  decrease  in  a  given  number  of  swings,  and  thus  to 
allow  for  the  effect  of  resistances  in  diminishing  the  first  swing 
from  its  true  value  to  what  is  observed. 

(2)  The  discharge  of  a  leyden  jar,  for  the  current  is  proportional 
to  the  remaining  charge  which  drives  it. 

(3)  The  equalization  of  temperature  by  conduction,  for  the  flow 
of  heat  is  proportional  at  each  instant  to  the  outstanding  difference 
of  temperature. 

(4)  The  progress  of  a  chemical  reaction  between  two  substances, 
for  the  rate  of  change  at  each  stage  depends  on  the  amount  of 
uncombined  reagents  left. 


EXAMPLES. 

1.  A  block  of  iron  weighing  561bs.  rests  on  a  stone  floor,  the  coefficient 
of  friction  being  -3.     What  is  the  least  force  that  will  move  it  if  applied 
(1)  horizontally,  (2)  at  45°  to  the  horizontal? 

2.  The  base  of  an  inclined  plane  is  4  feet  and  the  height  3  feet.     A  force 
of  8  Ibs.  weight  parallel  to  the  plane  will  just  prevent  a  mass  of  20  Ibs.  from 
sliding  down.     Find  the  coefficient  of  friction. 


204  MECHANICS  [CHAP  xix 

3.  The  weight  on  the  driving  wheels  of  a  locomotive  is  30  tons.     What  is 
the  greatest  pull  the  engine  can  exert  if  the  coefficient  for  iron  on  iron  is  -2  ? 

4.  A  weight  of  60  Ibs.  is  on  the  point  of  motion  down  a  rough  inclined 
plane  when  supported  by  a  force  of  24  Ibs.  weight  acting  up  the  plane ;  and 
is  on  the  point  of  motion  up  the  plane  when  the  force  is  increased  to  36  Ibs. 
weight.     Find  the  coefficient  of  friction  and  the  inclination  of  the  plane. 

5.  A  mass  of  80  Ibs.  rests  on  a  plane  (/x  =  *15)  inclined  30°  to  the  horizon. 
Compare  the  forces  required  to  draw  it  up  the  plane  by  a  rope  parallel  to  the 
plane  and  by  a  horizontal  push. 

6.  A  heavy  beam  rests  with  one  end  on  a  rough  floor  and  the  other 
against  a  rough  vertical  wall,  and  is  on  the  point  of  slipping  when  inclined 
30°  to  the  horizon.     Find  /z,  supposed  to  be  the  same  for  both  surfaces. 

7.  A  ladder  20  feet  long  and  weighing  70  Ibs.  rests  at  45°  against 
a  rough  vertical  wall.     A  man  weighing   10  stone  climbs  up  it.     If  the 
coefficient  of  friction  at  each  end  is  -5,  shew  that  it  will  slip  when  he  has 
gone  up  13  feet. 

8.  Three  complete  turns  of  a  rope  are  taken  round  a  rough  post,  and 
one  end  of  the  rope  is  held  with  a  force  of  100  Ibs.  weight.    What  force  would 
be  required  at  the  other  end  in  order  to  make  it  slip,  if  the  coefficient  of 
friction  is  '3? 


BOOK  III. 

APPLICATION   TO   VAEIOUS  PKOBLEMS. 


CHAPTER  XX. 

MOTION   ON  AN   INCLINED   PLANE.    BRACHISTOCHRONES. 

193.     To  find  the  motion  of  a  body  on  a  smooth  inclined 
plane. 

R 


The  weight  W  may  be  resolved  into  components  W  sin  t,  down 
the  plane,  and  Wcosi  perpendicular  to  it.  Since  there  is  no 
motion,  and  therefore  no  acceleration,  perpendicular  to  the  plane, 
the  latter  component  must  be  just  balanced  by  the  pressure  of  the 
plane. 

The  weight  W  would  produce  an  acceleration  g  in  the 
body.  The  component  W  sin  i  produces  the  acceleration  g  sin  i 
downwards.  This  value  must  be  substituted  for  g  in  the 


208 


MECHANICS 


[CHAP. 


kinematic   formulae    of  §  111.     The  velocity  acquired   and   the 
distance  travelled  in  any  time  t  from  rest  are  given  by 

v  =  g  sin  i  .  t, 

' 


Also,  if  the  body  starts  from  rest  at  B,  and  v  is  its  velocity  when 
it  arrives  at  A,  then 

V2=  2g  sin  i.  AB 

=  2gAB .  sin  i 
=  2g.BC. 

Thus  the  velocity  at  A  is  that  due  to  the  vertical  fall  BC, 
whatever  the  slope  of  the  plane.     (§  66.) 

194.     The  time  of  descent  down  all  chords  of  a  vertical  circle 
terminating  in  the  highest  or  lowest  points  is  the  same. 

Let  t  be  the  time  of  descent  down  A 

AC.     The  acceleration  is  g  cos  0. 

Then 


and  <= 


2A.O 


-v 


ZABcosd 
gcosO 


/2AB 

' 


The  time  is  obviously  the  same  down 
the  corresponding  chord  G'B  ending 
in  B. 

195.  To  find  the  shortest  time  of  descent  by  an  inclined  plane 
from  a  given  point  to  a  given  vertical  circle. 

Such  paths  of  shortest  time  are  called  brachistochrones. 

Could  a  circle  be  drawn  to  touch  the  given  circle  AB,  and 
have  its  highest  point  at  the  given  point  P,  the  problem  would  be 
solved.  For  the  time  down  PQ  to  Q,  the  point  of  contact,  is 
obviously  less  than  that  down  any  other  chord  PQ'R'. 

To  describe  the  circle,  join  P  to  the  lowest  point  of  AB.  This 
gives  the  path.  For  draw  the  vertical  at  P,  and  let  the  radius 
CQ  produced  cut  it  in  0.  Then 

Z  OQP  =  z  CQB  =  z  CBQ  =  z  QPO.    .'.  OQ  =  OP. 


XX]  BRACHISTOCHRONES 

Hence  0  is  the  centre  of  the  circle  required. 


209 


Fig.  98. 

Similar  problems  about  circles  are  generally  solved  by  joining 
a  point  to  the  highest  or  lowest  points  of  the  circles. 

196.     The  path  of  quickest  descent  between  two  curves  bisects 


...-•-'  R' 


Fig.  99. 


C. 


14 


210  MECHANICS  [CHAP. 

the  angle  between  the  vertical  arid  the  normal  to  the  curve  at 
each  end. 

If  PQ  is  the  path,  P  must  be  the  highest  point  of  a  circle 
touching  the  lower  curve  at  Q,  OQ  must  be  normal  to  the  curve, 
and  PQ  obviously  bisects  OQV. 

Similarly  Q  must  be  the  lowest  point  of  a  circle  touching  the 
upper  curve  at  P,  and  the  property  is  true  for  that  end  also. 

197.  A  curve  may  be  regarded  as  the  limiting  form  of  a  very 
Motion  on  a  curve     large   number   of 

under  gravity.          inclined      planes, 

joined  end  to  end,  when  the 
number  of  planes  is  made  infi- 
nitely great,  and  each  plane 
infinitely  short. 

A  body  sliding  down  the  curve 

acquires  in  each  element  of  its  path  through  the  action  of  its 
weight  the  extra  velocity  due  to  the  vertical  height  fallen 
through;  and  in  the  limit  the  pressure  from  the  curve,  which 
changes  the  direction  of  motion,  is  everywhere  at  right  angles  to 
the  tangent,  so  that  it  can  do  no  work  which  could  change  the 
magnitude  of  the  velocity  along  the  curve  ;  so  that  the  same  law 
holds  good  for  a  finite  length  of  the  curve.  The  velocity  acquired 
in  sliding  from  A  to  P  is  the  same  as  that  acquired  in  falling  from 
the  vertical  height  BP.  Galileo  perceived  this  principle  (§  66), 
which  is  indeed  the  principle  of  Work.  If  m  be  the  mass  of  the 
body,  v  the  velocity  at  P,  h  the  vertical  height  fallen  through, 

the  energy  rav2/2  =W.h  (the  work  done  by  the  weight)  =  mgh, 
and  fl2  =  2gh. 

198.  Let  a  particle  P  be  projected  horizontally  from  A  with 
Motion  on  a        velocity  F,  on  a  smooth  circle  APA'.     To  find 

circle.     ^e  pressure  on  the  curve  at  any  point. 

Let  a  be  the  radius  of  the  circle  ;  A  OP  =  6. 
(1)    The  velocity  v  at  P  is  given  by 


F2 

-cos  0). 


XX] 


MOTION   ON  A   VERTICAL   CIRCLE 


211 


(2)  At  P  the  particle  is  describ- 
ing a  circle  of  radius  a  with  velocity 
v.  Its  acceleration  along  the  radius 

2 

PO  is  -  (§  77),  depending  only  on 

Cb 

the  velocity  at  the  moment. 

The  resultant  force  acting  on  it 
along  PO  must  be  such  as  to  give 
it  this  acceleration.  The  forces  re- 
solved along  PO  are  W  cos  6  —  R, 
where  H  is  the  outward  pressure  of 
the  circle. 

v2      Wcos0-R 


and 


a  m 

R=W  COS0-  — 
a 

q 
=  mg  cos  9  —  - 


Fig.  101. 


(  ya) 

=  m  ]  (30  cos  0-20)  --  L 
(.  a) 

If  the  particle  slide  from  rest,  F=  0  and 


This  vanishes,  i.e.  the  particle  will  leave  the  circle,  when 

3  cos  6  -  2  =  0, 
or  cos  0  =    . 


EXAMPLES. 

1.  A  body  is  projected  up  a  smooth  plane  inclined  30°  to  the  horizon  with 
a  velocity  of  80  feet  per  second.     Find  how  far  up  the  plane  it  will  go  before 
coming  to  rest,  and  the  time  occupied  before  it  reaches  the  bottom  again. 

2.  A  toboggan  on  iron  runners  slides  down  an  ice  slope  of  1  in  8  for  a 
distance  of  100  yards.     Find  the  time  occupied,  and  the  speed  at  the  bottom 
neglecting  friction. 

14—2 


212  MECHANICS  [CHAP,  xx 

3.  Work  question  2  allowing  for  a  friction  of  /z  =  '05  ;  and  find  how  far 
the  toboggan  will  then  run  on  the  level. 

4.  A  heavy  pendulum  bob  suspended  by  a  wire  5  feet  long  is  drawn  aside 
3  feet  from  the  vertical  and  then  let  go.    Find  the  velocity  at  the  lowest  point 
of  the  swing. 

5.  A  boy  on  a  swing  20  feet  long  works  himself  up  till  he  can  just  touch 
with  his  feet  a  beam  4  feet  below  that  from  which  the  swing  is  hung.     Shew 
that  he  must  be  going  more  than  20  miles  an  hour  at  the  lowest  point. 

6.  Give  a  geometrical  construction  for  the  line  of  quickest  descent  from 
a  given  point  to  a  given  inclined  plane. 

7.  Find  the  shortest  time  in  which  a  ring  can  be  made  to  slide  down 
a  wire  to  a  vertical  wall  from  a  point  distant  20  feet  from  it. 

8.  Find  the  inclination  of  a  plane  down  which  a  particle  slides  through 
a  vertical  height  of  1  foot  in  half  a  second. 

9.  A  one-ounce  bullet,  hung  by  a  string  4  feet  long  from  a  fixed  point,  is 
projected  so  as  just  to  describe  a  vertical  circle  without  allowing  the  string  to 
slacken  when  the  bullet  is  at  the  top.   Find  the  tension  of  the  string  when  the 
bullet  is  at  the  bottom. 

10.  In  a  centrifugal  railway  the  car  runs  down  a  long  slope  and  then  the 
rails  take  a  turn  round  a  vertical  circle  of  30  feet  diameter,  so  that  the  car  and 
passengers  are  upside  down  at  the  top.     What  is  the  least  vertical  height 
through  which  the  car  must  descend  before  entering  the  circle  in  order  that 
it  may  not  leave  the  rails  ? 

11.  A  man  is  caught  by  the  sail  of  a  windmill,  which  is  29  feet  long  and 
revolves  10  times  a  minute.     Shew  that  for  a  moment,  just  as  he  passes  the 
top,  he  might  let  go  his  hold  without  falling. 


CHAPTER  XXL 


PEOJECTILES. 


199.  GALILEO  shewed  (§  74) 
that  the  path  of  a  projectile  in 
vacuo  is  a  parabola. 

One  way  of  verifying  this  ex- 
perimentally, when  the  particle  is 
projected  horizontally,  is  to  allow 
a  stream  of  water  to  issue  in  a 
horizontal  jet  from  a  supply  at  a 
steady  pressure.  The  shadow  of 
such  a  jet  may  be  thrown  by  a 
distant  candle  on  a  sheet  of  ground 
glass,  and  traced  with  a  pencil. 
The  curve  obtained  will  prove  to 
be  a  parabola. 

We  shall  describe  a  method 
devised  by  Morin  to  shew  that 
if  the  motion  of  a  body  falling 
freely  is  compounded  with  a 
uniform  horizontal  velocity,  the 
result  is  a  parabola. 

Experiment.  Morin's  Machine. 

A  sheet  of  paper  is  wrapped 
round  a  vertical  cylinder  which  is 
turned  uniformly  on  its  axis  by 
hand  or  by  a  motor.  A  small 
weight  slides  on  a  vertical  rod  at 
the  side,  and  carries  with  it  a 
pencil  which  presses  against  the 
paper.  While  the  weight  is  held 


Kg.  102. 


214  MECHANICS  [CHAP. 

at  the  top  of  the  rod,  the  pencil  traces  a  horizontal  circle  on  the 
paper  with  uniform  speed.  But  when  it  is  allowed  to  fall,  it 
traces  a  curve  resulting  from  the  combination  of  the  uniform 
horizontal  motion  of  the  paper  with  the  motion  of  the  freely 
falling  weight.  This  curve  proves  to  be  a  parabola. 

The  arrangement  suffers  from  two  defects.  It  is  not  easy  to 
maintain  absolute  uniformity  in  turning  the  cylinder;  and  the 
friction  of  the  pencil  on  the  paper,  and  of  the  small  weight  on  the 
rod,  seriously  affects  the  free  movement  of  the  weight.  These 
defects  are  overcome  in  a  form  of  machine  designed  by  the 
Cambridge  Scientific  Instrument  Company. 

The  pencil  is  in  this  case  fixed  on  a  vertical  rod  and  can  be 
adjusted  so  as  to  press  against  the  paper  at  the  bottom  of  the 
cylinder.  The  cylinder  itself  falls  after  being  set  in  rapid  rotation. 
It  may  be  made  very  heavy,  and  yet  brought  to  rest  conveniently 
by  a  leaky  piston  arrangement  at  the  end  of  the  fall. 

The  rotation  of  the  heavy  cylinder  is  practically  uniform 
throughout  the  short  time  of  fall  (cf.  Galileo's  Principle,  §  70), 
since  the  very  small  frictions  on  the  guiding  rod  and  against  the 
pencil  are  inconsiderable  compared  with  the  momentum  of  the 
heavy  rotating  cylinder ;  and  if  the  rod  is  made  truly  vertical  by 
means  of  the  levelling  screws,  and  is  well  oiled,  the  fall  is  practically 
"  free  "  for  the  same  reasons. 

When  the  paper  is  unwrapped,  a  curve  is  obtained  such  as 
Fig.  104,  inverted  in  the  case  of  the  second  machine. 

The  only  difficulty  is  to  decide  on  the  exact  point  A  of  the 
horizontal  line,  corresponding  to  the  moment  of  release. 

Choose  a  point  A,  and  with  a  set  square  draw  ANN'  vertical, 
and  mark  off  inches  along  it.  At  two  divisions  Nt  N',  say  3  and 
6  inches  from  A,  draw  horizontal  lines  NP,  N'P'  to  the  curve,  and 
measure  their  lengths  carefully.  Then  we  should  find 

PN*     PN'* 
AN      AN'  ' 

If  this  is  not  very  approximately  correct,  let  us  assume  that 
the  axis  should  have  been  drawn  a  short  distance  x  to  the  left,  as 
A'nri.  Then  we  should  have 

(PN  +  x?  _(P'N'  +  x)* 
A'n       ~  "    An 


XXI] 


MORINS   MACHINE 


215 


A'A 


ri 


Fig.  104. 


Fig.  103. 


216 


MECHANICS 


[CHAP. 


Solving,  we  find  how  far  to  the  left  the  axis  should  have  been  drawn. 
If  x  turns  out  negative,  the  axis  should  be  drawn  to  the  right  of 
AN.  Having  drawn  the  new  axis,  verify  for  several  other  dis- 
tances AN"  that  PNZIAN  is  constant. 

Let  v  be  the  horizontal  velocity  of  the  paper,  t  the  time  of 
fall. 

Then  PN  =  vt;  AN  =  gt*/2.  So  that  PN^jAN  has  the  value 
2v*/g.  We  may  employ  this  result  either  to  find  the  speed  at 
which  the  paper  was  moving  past  the  pencil  horizontally,  if  we 
assume  g  =  32'2  ;  or  to  determine  g  if  we  have  other  means  of 
finding  the  speed  of  the  paper,  as,  for  instance,  by  substituting  for 
the  pencil  a  tuning  fork  carrying  a  pointer  after  the  manner  of 
§  263  and  using  smoked  paper. 

200.     The  proof  of  §  74  may  be  generalized  for  oblique  pro- 

Oblique  Pro-      jectlOn. 

jection.  Let  a  k0(}y  be  projected  in  the  direction  PT  with 

velocity  u. 

If  gravity   ceased   to   act,   it   would   travel   along   PT  with 
uniform   velocity,  and   at 
the  end  of  t  seconds  would 
arrive  at  T  where 


=  ut. 

If,  on  the  other  hand, 
it  were  merely  dropped  at 
P,  it  would  in  the  same 
time  t  reach  V  where 

PF-f. 


Fig.  105. 


The  initial  velocity  and 
gravity  together  bring  it  to  Q,  the  other  corner  of  the  parallelogram 
on  PT,  PV. 

By  taking  different  times,  t,  we  may  construct  any  number  of 
points  Q  on  its  path. 

Draw  the  vertical  PM,  and  make  PM  =  ^-  . 

ty 
On  the  other  side  of  PT  make    tTPS=  Z  TPM,  and   take 


XXI] 


PARABOLIC   MOTION 


217 


PS  =  PM.  •   Draw  ML  horizontal,  and  with  focus  8  and  directrix 
SL  describe  a  parabola.     This  is  the  path. 

It  passes  through  P,  for  SP  =  PM ;  and  PT  is  a  tangent,  for 
it  bisects  8PM ;  PV  is  a  diameter. 

Also  QVz  =  PT*=uHz  =  --.^  =  4SP.PV. 

Hence  Q  is  on  the  parabola,  and  so  is  every  other  point  similarly 
constructed. 

201.  The  velocity  at  any  point  in  the  parabola  is  that  due  to 
a  free  fall  from  the  directrix  to  that  point. 

For  any  point  may  be  regarded  as  the  point  of  projection.  But 
the  velocity  due  to  a  fall  through  HP  is  given  by 

%-g.MP  (§111), 


so  that  v  =  u,  the  velocity  at  P. 

202.     To  find  the  "elevation,"  or  angle  of  projection,  for  a 
given  velocity  in  order  to  strike  a  given  point. 


fig.  106. 


218  MECHANICS  [CHAP, 

Let  the  body  be  projected  from  P  with  velocity  F;  let  Q  be 
the  point  to  be  struck. 

Draw  PM  vertical  and  equal  to  F2/2#.  The  horizontal  MN 
will  be  the  directrix,  and  P,  Q  are  both  on  the  parabola.  Draw 
QN  vertical,  and  describe  the  circles  MS&,  NS&  having  P  and  Q 
as  centres.  Either  point  of  intersection  will  do  for  a  focus. 
Bisect  the  angles  MPS1}  MPSa.  These  are  the  directions  of  pro- 
jection. 

There  are  thus  in  general  two  possible  angles  of  elevation  for 
hitting  a  given  object  with  a  given  velocity  of  projection.  These 
reduce  to  one  if  the  circles  touch ;  if  they  do  not  intersect,  the 
velocity  is  not  great  enough  to  reach  Q. 


203.  The  parabola  helps  us  to  realize  the  path  as  a  whole, 
Analytical  and  problems  may  often  be  neatly  solved  by 
Method.  means  of  its  properties.  But  it  is  usually  better 

to  resolve  the  motion,  and  consider  the  horizontal  and  vertical 

components  separately. 

Let  the  body  be  projected  with  velocity  u  in  a  direction  making 

an  angle  a  with  the  horizontal. 

Resolve  u  into  a  horizontal  component  ucosa,  and  a  vertical 

component  u  sin  a. 

The  only  force  acting  on  the  body,  once  it  is  projected,  is  the 

weight,  which  causes  a  vertical  acceleration  g  downwards.     Thus 

we  have 

Initial  velocity.        Acceleration. 

for  the  horizontal  motion u  cos  a,  0 

„         vertical  „  wsina,  —  g 

(1)     To  find  the  velocity  after  t  seconds. 

The  horizontal  component  =  u  cos  a (1). 

(There  is  no  acceleration  to  change  it.  Hence  the  horizontal 
velocity  is  constant ;  in  order  to  keep  exactly  underneath  a  ball 
a  cricketer  must  run  with  uniform  speed.) 

The  vertical  component  =  wsin  a  —gt (2). 


XXI]  PARABOLIC   MOTION  219 

If  v  is  the  resultant  velocity,  inclined  6  to  the  horizon, 


cos2  a  +  (u  sin  a  — 


„      u  sin  a  —  at  ' 

tan  6  =  -         —2-  .  •£> 

u  cos  a  i 

s 

(2)     To    find    the    position    at      -| 
time  t. 


Horizontal  distance  ==  u  cos  a  .  t. 


* 


Vertical  height  =  u  sin  a  .  t  —  ¥--  .  Fig.  107. 

(3)     To  find  T  the  time  of  flight. 

Gravity  destroys  a  vertical  velocity  of  g  feet  per  second  in 
every  second.  The  body  will  be  at  the  top  of  its  path  when  gravity 
has  just  destroyed  its  upward  velocity  u  sin  a  ;  i.e.  in  u  sin  a/g 
seconds.  It  takes  as  long  to  come  down.  So  the  time  of  flight 

_  2u  sin  a 

9 

(Or,  by  formula  (2),  at  the  top  u  sin  a  —  gt  =  0  ;  hence 
u  sin  a  f    _,  _  2u  sin  a 

*  ==    -    5        JL    -    --     • 

9  9 

Again,  by  formula  (4),  at  the  ground 

u  sin  a  .  t  —  ^-  =  0, 


The  first  value  corresponds  to  the  start,  the  second  to  the  finish.) 

(4)     To  find  the  range. 

During  the  whole  time  of  flight  the  horizontal  velocity  is  con- 
stant. Hence  the  distance  of  the  point  where  the  projectile 
strikes 

=  time  of  flight  x  horizontal  velocity 

2u  sin  a 


9 

it?  sin  2  a 


x  u  cos  a 


220  MECHANICS  [CHAP. 

For   a   given   velocity    of    projection    this  is   greatest   when 
sin  2a  =  1 ;  i.e.  when  2a  =  90°,  and  a  =  45°. 

(5)     To  find  the  greatest  height  reached. 

This  is  the  same  as  for  a  body  shot  vertically  upwards  with 
velocity  ^sin  a.     Let  h  be  the  height.     Then  (§  111), 

u2  sin2  a 


,      w2  sn2  a 
and  h  =  —  =  -- 


(Or,  by  formula  (4),  height  at  time  - 

t/ 

u  sin  a      or  w2  sin2  a 
=  M-  sm  a  .  ---  §  •  — 

2         * 


(6)     To  find  the  Latus  Rectum  of  the  parabola  described. 

At  the  top  the  vertical  velocity  has  vanished,  and  only  the 
horizontal  component  u  cos  a  is  left.  But  this  is  the  velocity  due 
to  a  fall  from  the  directrix  to  the  vertex,  i.e.  through  one  quarter  of 
the  latus  rectum.  Hence  the  latus  rectum 


w2  cos2  a 
4  x  —  -  - 


cos2  a 


204.     Problems  relating  to  projection  above  an  inclined  plane 
Projection  over  an     are  ^es^  solved  by  resolving  along  and  perpen- 
dicular to   the   plane.     Note   the   method   only, 
which  will  be  seen  from  a  couple  of  examples. 


Fig.  108. 


XXI]  PARABOLIC   MOTION  221 

Ex.  1.     A  body  is  projected  at  elevation  a  with  velocity  u, 
from  the  foot  of  a  hill  of  slope  i\  find  the  range  on  the  hill. 
Here  the  initial  velocity  has  components 

u  cos  (a  —  i)  along  the  hill, 

and  u  sin  (a  —  i)  perpendicular  to  it. 

The  acceleration  of  gravity  has  components 

—  gsini  up  the  hill, 

and  —gcosi  perpendicular  to  it. 

Velocity  perpendicular  to  the  hill  at  time  t  is 
u  sin  (a  —  i)—g  cos  i .  t. 

This  vanishes  when 

u  sin  (a  —  i) 

t  = ; , 

g  cos  i 
and  the  time  of  flight 

_  2u  sin  (OL  —  i) 
g  cosi 

Distance  up  the  hill  at  time  t 

a  sin  i 

=  ucos(a-i).t-  ?—= — .  t\ 
2t 

This  will  be  the  range  when 

2w  sin  (a  —  i) 
g  cos  i 

.    -D          _  2/^2  sin  (a  —  i)  cos  (a  —  i)     g  sin  i  4w2  sin2  (a  —  i) 
g  cos  i  2     '       g*  cos2  * 


sin  (a  - 


-  *)  (  sin  ^ .  sin  (a  —  ^)) 

-. — -^cos(a-i) — ^ '\ 

>'        (  cos^         J 


_  2w2  sin  (a  —  i)  cos  a 
#  '          cos2i 

Of  course  when  i  =  0  this  reduces  to  the  range  on  a  horizontal 
plane 

w2  sin  2a ' 

ff 

Ex.  2.     To  find  the  condition  that  the  projectile  should  strike 
the  hill  at  right  angles. 


222  MECHANICS  [CHAP. 

The  velocity  parallel  to  the  hill  at  the  moment  of  striking 


,   ,  .  2^sm(a  —  t) 

must  be  zero  :   i.e.,  at  t  =  —        -  —  :  —  -  ,  we  must  have 

gcosi 

u  cos  (a.  —  i)—g  sin  i  .  t  =  0, 

.     .  2%  sin  (a  —  i) 

.'.  u  cos  (a  —  i)  —  g  sim  .  -  =0. 

gcosi 

.'.   tan  i  =  J  cot  (a  -  i),     . 
which  is  the  condition  required. 


EXAMPLES. 

1.  A  ball  is  projected  with  a  velocity  of  80  feet  per  second  at  an 
elevation  of  30° ;  find  the  time  of  flight,  range,  and  greatest  height  attained. 

2.  The  muzzle  velocity  of  a  gun  is  2240  feet  per  second.   Find  its  greatest 
range  in  miles. 

3.  A  shot  is  fired  horizontally  from  a  height  of  4  feet  above  the  ground, 
and  strikes  the  ground  at  a  distance  of  1000  feet.     What  was  its  initial 
velocity  ? 

4.  A  man  can  throw  a  stone  with  a  velocity  of  80  feet  per  second.     In 
what  direction  must  he  throw  it  so  as  to  strike  a  bird  20  feet  above  his 
shoulder  on  a  tree  30  feet  away  from  him  ? 

5.  A  shot  is  fired  from  a  fort  144  feet  above  the  sea  with  a  velocity  of 
2000  feet  per  second  at  an  elevation  of  30°.    At  what  horizontal  distance  from 
the  fort  will  it  strike  the  water  ? 

6.  The  record  throw  for  the  cricket  ball  is  127  yards  1  foot  3  inches. 
Find  the  least  initial  velocity  the  ball  could  have  received. 

7.  The  record  for  the  16-lb.  hammer  is  171  feet.     If  the  handle  is  4  feet 
long,  find  how  many  times  per  minute  it  was  being  whirled  round,  assuming 
that  it  was  let  go  in  the  most  favourable  position. 

8.  The  back  lines  of  a  tennis  court  are  78  feet  apart,  and  the  service 
lines  42  feet.     The  net  is  3  ft.  3  in.  high. 

Find  the  horizontal  velocity  of  the  ball 

(1)  when  it  is  returned  from  near  the  ground  at  one  back  line  so  as  to 
graze  the  net  and  just  strike  the  other  back  line  ; 

(2)  when  it  is  served  from  a  height  of  8  feet,  grazes  the  net,  and  strikes 
the  service  line. 


XXI]  EXAMPLES  223 

9.  A  bullet  is  fired  from  a  rifle  4  ft.  above  the  ground  with  a  velocity  of 
2000  feet  per  second,  so  as  to  strike  a  target  at  500  yards  at  the  same 
height  of  4  feet.     Shew  that  the  rifle  must  be  sighted  for  an  elevation  of 
about  20' ;  and  that  a  man  6  ft.  2  in.  in  height  could  stand  halfway  between 
the  rifle  and  target  without  being  hit. 

10.  The  record  for  the  high  jump  is  6  ft.  5|  in.     What  is  the  greatest 
upward  velocity  a  man  can  give  his  body  in  leaping,  assuming  that  his  centre 
of  gravity  is  3  ft  6  in.  above  the  ground  and  that  it  rose  15  inches  above  the 
bar? 

What  height  could  the  same  man  jump  at  the  surface  of  the  moon,  where 
gravity  is  150  c.G.s.  units  ? 

11.  A  shot  is  projected  with  a  velocity  of  2000  feet  per  second  at  an 
angle  of  30°  with  a  hill  which  is  itself  inclined  30°  to  the  horizon.     Find  the 
time  of  flight  and  range  up  the  hill. 

12.  The  greatest  range  of  a  gun  on  a  horizontal  plane  is  3000  feet.    Shew 
that  its  greatest  ranges  up  and  down  a  plane  inclined  30°  to  the  horizon  are 
2000  feet  and  2000  yards  respectively. 

13.  The  record  for  putting  the  weight  (16  Ibs.)  is  48  ft.  2  in.     Assuming 
that  it  is  let  go  at  a  height  of  6  feet  above  the  ground,  and  elevation  45°, 
calculate  the  speed  with  which  it  is  projected.     If  in  the  act  of  projection  it 
is  heaved  up  through  a  vertical  height  of  2  feet,  find  the  foot-pounds  of  work 
put  into  it. 

14.  The  Norwegian  Ski-jumping  contests  in  February,  1904,  took  place 
on  a  snow  slope  at  Holmenkollen,  186  yards  long.     The  competitors  slid 
down  two-thirds  of  the  slope  (which  was  in  this  part  inclined  15°  to  the 
horizon)  to  a  ledge,  from  which  they  took  off  for  the  jump.     Below  the  ledge 
the  steepness  of  the  slope  increased  to  24°.     Supposing  that  the  coefficient  of 
friction  between  the  ski  and  the  ice  was  -05,  and  that  the  lip  of  the  ledge  was 
so  curved  as  to  give  the  jumper  an  elevation  of  6°  above  the  horizon  at  the 
take-off,  shew  that  the  speed  at  the  ledge  would  be  about  48  miles  an  hour, 
and  the  leap  about  104  feet. 


CHAPTER  XXII 


SIMPLE   HARMONIC  MOTION. 

205.  Definition.  Let  a  point  P  describe  a  circle  with  constant 
velocity.  Draw  PN  perpendicular  to  the  diameter  AOA'.  The 
motion  of  N  is  called  a  Simple  Harmonic  Motion  (S.H.M.). 

Let  the  diameter  AOA'  be  horizontal. 

As  P  revolves  constantly  round  the  circle,  N  vibrates  or 
oscillates  about  the  centre  0,  be- 
tween the  extreme  positions  A  and 
A'.  At  A  the  motion  of  P  is 
entirely  vertical,  and  N  is  for  an 
instant  at  rest.  It  then  gains 
speed  towards  0,  until  at  0  it  has 
momentarily  the  full  speed  of  P, 
moving  parallel  to  AOA'  at  B 
overhead.  Next  N  slackens  pace 
till  it  comes  to  rest  for  an  instant 
at  A' ;  reverses  its  motion ;  and  at 
0  has  the  full  speed  of  P  to  the 
right;  and  finally  slackens  again 
till  it  comes  to  rest  at  A  as  P 


Fig.  109. 


passes  through.     The  motion   is  repeated  with  every  revolution 
of  P. 

This  kind  of  motion  is  extremely  important.  It  may  almost 
be  said  to  share  the  whole  range  of  Physics  with  the  law  of 
gravitation;  for  whenever  we  are  not  dealing  with  the  mutual 
action  of  distant  bodies  by  means  of  the  one,  we  shall  almost 


UNIVERSITY 

OF 


CHAP.  XXII] 


THE  HODOGRAPH 


225 


certainly  find  ourselves  working  out  the  movements  of  connected 
bodies  or  parts  of  bodies  by  the  other. 

The  simplest  way  to  study  it  is  to  observe  that  the  point  N  is 
subject  to  the  horizontal  motion  of  P  without  any  of  its  vertical 
motion.  We  know  the  velocity  of  P;  and  its  acceleration  has 
been  found  (§  77).  The  velocity  and  acceleration  of  N  at  any 
instant  will  be  the  horizontal  components  of  the  velocity  and 
acceleration  of  P  at  that  instant. 

But  first  we  will  give  another  investigation  of  the  acceleration 
of  P  partly  for  the  sake  of  the  importance  of  the  result ;  partly, 
too,  for  the  interest  of  the  graphical  method  it  illustrates,  that  of 
the  Hodograph. 


Fig.  110. 


206. 


The  Hodograph. 


Definition.  Let  a  point  describe  any  curve  PQR  in  any 
manner.  Make  a  construction  diagram  of  its 
velocity.  From  any  point  0  draw  Op  to  represent 
in  magnitude  and  direction  its  velocity  at  P ;  Oq  to  represent  its 
velocity  at  Q,  and  so  on.  For  successive  points  on  PQR  there  will 
be  corresponding  points  pqr  through  which  a  curve  may  be  drawn. 
This  curve  is  called  the  Hodograph  of  the  orbit  PQR. 
Property  of  the  Hodograph. 

The  velocity  in  the  hodograph  represents  the  acceleration  at  the 
corresponding  point  of  the  orbit. 

For,  by  the  triangle  of  velocities,  the  straight  line  pq  represents 
ie  velocity  which  must  be  compounded  with  Op  to  produce  Oq ; 
?.,  pq  is  the  velocity  gained  by  the  point  in  passing  from  P  to  Q 
in  the  orbit,  while,  if  q  is  taken  very  near  to  p,  the  line  pq  becomes 
the  arc  of  the  hodograph. 

c.  15 


226 


MECHANICS 


[CHAP. 


The  rate  at  which  pq  is  described  is  thus  the  rate  of  change  of 
velocity  in  the  orbit,  and  also  the  rate  of  description  of  arc  in  the 
hodograph. 

Hence  the  velocity  in  the  hodograph  is  the  acceleration  in  the 
orbit. 

The  advantage  of  the  hodograph  is  that  it  reduces  the  study 
of  accelerations  to  that  of  velocities,  usually  less  complex. 


207.     Let  P  describe  a  circle,  centre  0,  radius  OP  =  r,  with 
velocity  v. 

Acceleration  of  a  * 

point  which  Take    0    for 

describes  a  circle  .    .  _     .        .. 

with  uniform  Origin    OI   the   ho- 

dograph.  Draw 
Op  representing  v,  the  velocity 
at  P,  and  therefore  parallel  to 
the  tangent  PT.  As  P  moves 
round  0,  the  line  Op  revolves 
with  constant  length.  Thus  the 
hodograph  is  a  circle,  and  the 
velocity  of  p  is  at  right  angles  to 
Op,  i.e.,  parallel  to  PO. 

The  acceleration  of  P  is  there- 
fore along  PO,  i.e.,  towards  the 
centre.  And  since  the  circles  are 
described  in  equal  times, 


Fig.  ill. 


Velocity  of  p      radius  Op 
Velocity  of  P     radius  OP  * 

Acceleration  of  P     v 


and  acceleration  of  P  =  — . 

r 

As  in  §  77,  we  may  write 

acceleration  of  P 


4-7T2 


where  T  is  the  time  of  one  complete  revolution,  or  Periodic  Time 
of  P. 


XXII] 


SIMPLE    HARMONIC   MOTION 


227 


208.     To  find  the  Acceleration  in  a  S.H.M. 
Let    T    be    the    periodic    time 
of  P. 

4?r2 
The  acceleration  of  P  is   ^  OP 

along  PO. 

The   acceleration   of  N=  hori- 
zontal  component   of   acceleration     A 
of  P 


Fig.  112. 


Note  that  (1)  the  acceleration  is  always  directed  towards  0,  for 
if  N  is  to  the  left  of  0,  ON  is  negative,  and  the  acceleration 
becomes  positive ;  and  (2)  its  magnitude  is  proportional  to  ON 
at  every  point,  for  the  coefficient  47r2/^2  is  the  same  throughout 
the  vibration.  This  is  no  longer  a  constant  acceleration,  like  that 
of  gravity  for  falling  bodies  and  projectiles,  but  an  acceleration 
proportional  to  the  distance  from  a  fixed  point,  and  always  directed 
towards  it. 

The  acceleration  is  greatest  at  A  and  A',  where  the  velocity  is 
zero  (but  is  being  reversed  in  direction) ;  and  vanishes  at  0,  where 
the  velocity  is  greatest  (and  for  a  moment  remains  equal  to  that 
of  P ;  hence  no  change  of  velocity,  i.e.,  no  acceleration). 

209.  The  force  required  to  produce  such  an  acceleration  in 
any  given  mass  must,  by  the  Second  Law  of  Motion,  be  proportional 
to  the  distance  of  the  mass  from  the  fixed  point  0.  This  is 
precisely  the  law  that  is  found  to  hold  good  when  elastic  bodies 
are  pulled  or  twisted  out  of  shape ;  and  in  general  for  any 
disturbance  of  stable  equilibrium  among  systems  of  material 
bodies,  or  the  parts  of  the  media  that  transmit  sound,  and  the 
phenomena  of  light,  electricity,  and  magnetism,  at  all  events  if 
the  disturbances  are  small.  The  forces  that  are  at  once  called 
into  play  tending  to  restore  the  normal  condition  of  equilibrium 

15—2 


228 


MECHANICS 


[CHAP. 


are,  for  small  disturbances,  proportional  to  the  displacements. 
Hence  the  various  systems  or  their  parts  begin  a  series  of  Simple 
Harmonic  Vibrations  about  their  positions  of  equilibrium.  Even 
if  the  initial  circumstances  lead  to  more  complicated  forms  of 
vibration,  Fourier  shewed  (Theorie  de  la  Chaleur)  how  to  resolve 
any  form  of  oscillation  into  a  series  of  Simple  Harmonic  Vi- 
brations. 


210.  1.  Suspend  a  small  scale 
Experimental  pan  (1)  by  an  elastic 
verification.  thread,  (2)  by  a  spiral 
spring,  in  front  of  a  vertical  scale, 
and  read  its  position.  Add  weights 
and  take  readings  of  the  position 
of  the  pan.  It  will  be  found  that 
the  amount  of  descent  is  propor- 
tional to  the  weight  added. 

The  principle  is  employed  in 
Joly's  Balance  for  small  weights, 
and  in  the  ordinary  spring  balance's 
for  large  ones. 

2.  Support  a  square  rod  of  iron 
or   wood    horizontally   across    two 
knife  edges  on  a  lathe  bed.    Attach 
a  vertical  millimeter  scale  to  the 
centre  of  the  rod,  and  hang  various 
weights  by  a  hook  at  the  centre. 
The  deflections  can  be  read  by  a 
fixed  microscope  furnished  with  a 
cross-hair,  and  directed  upon   the 
millimeter    scale.      They   will    be 
found   to   be   proportional   to  the 
weights  employed. 

3.  Suspend  a  heavy  cylinder, 
furnished  with   a   pointer   moving 
over    a    horizontal    circular    scale, 
by  a  brass  wire   hanging  from  a 


Fig.  113. 


XXII] 


SIMPLE   HARMONIC   MOTION 


229 


fixed  clamp.  Threads  attached  to  the  rim  of  the  cylinder,  and 
wound  round  it,  leave  it  at  opposite  ends  of  a  diameter,  and  passing 
over  pulleys  support  equal  small  weights.  The  angle  through 
which  the  cylinder  turns  may  be  read  by  graduations  on  the  rim, 
and  is  the  angle  through  which  the  wire  is  twisted.  Verify  that 
it  is  proportional  to  the  weight  used,  and  therefore  to  the  moment 
or  torque  of  the  twisting  couple. 

Thus  for  extension,  flexure  and  torsion  the  displacement  is 
proportional  to  the  force  required  to  produce  it;  so  that  the 
force  tending  to  restore  the  normal  condition  is  proportional  to 
the  displacement. 

211.     Def.  1.     The  greatest  distance  of  displacement  from  the 
position  of  equilibrium  is  called  the  Amplitude  of 

Amplitude — Pe-        i  .-i        ,  . 

riodicTime-       the  vibration. 

Def.  2.  The  time  of  one  complete  vibration,  i.e., 
between  two  passages  through  the  same  point  in  the  same  direc- 
tion, is  called  the  Period.  This  is  the  same  as  the  time  of  one 
revolution  of  P  in  the  circle.  The  period  or  periodic  time,  means 
the  time  of  a  double  swing,  say  from  A  to  A'  and  back  again. 
There  is  one  exception  to  this  rule;  in  all  work  relating  to  the 
measurement  of  time  it  has  become  the  custom  to  define  the 
"  seconds  "  pendulum  as  one  which  executes  a  half  oscillation  in 
one  second,  that  is  which  swings  from  A  to  A'  in  one  second. 

Def.  3.     The  position  of  the  point  N  in  its  vibration  is  called 
the  Phase.     It  is  measured  by 
the  fraction  of  a  vibration  which 
has  taken  place  since  the  be- 
ginning of  the  vibration. 

Let  t  be  the  time  elapsed 
since  the  beginning  of  the 
vibration  at  A\  6  the  angle 
described  by  OP.  Then  the 
phase  is  indicated  by  t/T  or 
0/27T.  Differences  of  phase  are 
often  expressed  as  differences  of 
the  angle  6,  directly  in  degrees. 

Def.   4.     If    the    time    is 


230  MECHANICS  [CHAP. 

measured,  not  from  the  moment  when  P  passes  through  A,  but 
from  some  other  position  of  P,  as  E,  then  the  angle  AOE  is  called 
the  Epoch,  and  is  generally  denoted  by  e. 

212.     Everything  about  a  S.H.M.  can  now  be  easily  expressed. 

Let  the  amplitude  (equal  to  the  radius  of  the  circle  of 
reference)  ber;  ON  =  x\  n  =  the  angular  velocity  of  OP  (more 
usually  denoted  by  o>  except  in  this  connection). 

Then  the  position,  velocity,  and  acceleration  of  N  are  given  by 

#  =  r  cos 6  =  T  cos  -fp  .t  =  r  cos nt, 
v  =  component  of  P's  velocity,  which  is  ^TrrfT, 
=  — „- .  r  sin  6  =  —  nr  sin  6  —  —  nr  sin  nb. 

a  =  — j- .  r  cos  6  =  —  ri*r  cos  6  —  —  ri*r  cos  nt 


If  the  time  is  measured  from  E,  we  have  only  to  add  the  epoch 
to  each  of  the  angles.  Thus  x  —  r  cos  (nt  +  e). 

213.     When  a  point  receives  displacements  (and  consequently 
velocities  and  accelerations)  which  are  at  every 

Composition  of  '         , 

simple  Harmonic  instant  the  resultants  of  those  due  to  two  S.H.M.  s, 
its  motion  is  said  to  be  compounded  of  the  two 
s.H.M.'s. 

In  1821  Fresnel  ("Memoire  sur  la  Diffraction,"  Comptes  Rendus, 
1826)  gave  a  rule  for  compounding  S.H.M.'s  of  the  same  period  and 
in  the  same  straight  line. 

Let  the  parallelogram  OACB  revolve  uniformly  round  the 
vertex  0,  and  drop  perpendiculars  AP,  BQ,  CR  on  a  fixed  line 
OX  Then  P,  Q,  R  all  describe  s.H.M.'s  along  OX.  Also 

OR  =  projection  of  00  on  OX 

=  sum  of  projections  of  OA,  AC 
=  sum  of  projections  of  OA,  OB 
=  OP  +  OQ. 


XXII] 


SIMPLE   HARMONIC   MOTION 


231 


Thus  the  displacement  of  R  is  always  the  sum  of  the  displace- 
ments of  P  and  Q.  Now  the  amplitudes  of  the  three  vibrations 
are  OA,  OB,  0(7;  and  their  phases  at  any  moment  are  fixed  by 
the  angles  XOA,  XOB,  XOC. 


Fig.  115. 

It  appears  that  the  resultant  of  two  S.H.M.'S  of  the  same  period 
and  in  the  same  straight  line  is  another  s.H.M.  in  that  line  with 
different  amplitude  and  phase,  ivhich  are  to  be  found  by  a  construction 
diagram  precisely  as  the  magnitude  and  direction  of  the  resultant 
of  two  forces  is  found  in  Statics. 

This  is  known  as  Fresnel's  Rule.  Thus,  from  any  point  0 
draw  OA  to  represent  the  amplitude  of  the  first  vibration  at  such 
an  angle  with  a  fixed  line  OX  as  will  represent  its  phase.  From 
A  draw  AC  to  represent  in  the  same  way  the  amplitude  and  phase 
of  the  second  vibration.  Join  0(7.  Then  OC  represents  the 
amplitude  and  phase  of  the  resultant  vibration. 

This  corresponds  to  the  Triangle  of  Forces  (§  166),  and  it  may 
obviously  be  extended  to  find  the  resultant  of  any  number  of 
S.H.M.'S,  as  in  the  Polygon  of  Forces  (§  168). 

The  result  is  extremely  important  in  the  theories  of  Sound, 
Light,  and  Electricity,  where  the  vibration  at  any  point  is  generally 
the  resultant  of  many  disturbances  received  simultaneously  from 
different  parts  of  an  advancing  wave. 


232 


MECHANICS 


[CHAP. 


Corollary  1.  If  the  component  vibrations  are  in  the  same 
phase,  OAC  is  a  straight  line,  and  the  amplitude  of  the  resultant 
is  the  sum  of  the  amplitudes  of  the  components.  If  they  differ 
by  half  a  period,  the  resultant  amplitude  is  the  difference  of  the 
amplitudes  of  the  components.  In  the  latter  case  if  the  amplitudes 
of  the  components  are  equal,  the  resultant  is  zero ;  so  that  two 
equal  S.H.M/S  differing  in  phase  by  half  a  period  destroy  each 
other. 

Corollary  2.  If  the  periods  are  very  nearly,  but  not  quite, 
equal,  one  vibration  will  gradually  gain  in  phase  on  the  other,  and 
the  effect  will  be  the  same  as  if  the  periods  were  exactly  equal, 
but  the  difference  of  phase  were  gradually  increased.  The  resultant 
will  be  a  vibration  with  changing  amplitude,  which  varies  between 
a  maximum,  equal  to  the  sum  of  the  original  amplitudes,  and  a 
minimum  equal  to  their  difference. 

Illustrations  of  this  effect  are  found  in  the  theories  of 
Interference  of  Sound  and  Light,  and  of  alternating  currents  in 
Electricity. 

214.  Two  s.H.M.'s  in  different  directions  generally  compound 
into  motion  in  a  curve.  By  way  of  example,  we  will  consider  two 
important  cases  of  the  composition  of  equal  S.H.M/S  of  the  same 
period  in  directions  at  right 
angles  to  each  other. 

(1)  When  the  phase  is  the 
same. 

Let  the  vibrations  start  from 
0  along  OA,  OB  at  the  same 
epoch.  Then  the  distances 
along  OA,  OB  are  the  same 
at  all  times  for  both.  A  point 
having  both  these  vibrations 
would  move  along  the  diagonal 
OA",  and  describe  a  s.H.M.  of 

the  same  period,  but  of  ampli-  Fig.  116. 

tude 

OA"  =  VOZ*  +  OB\ 


XXTl] 


SIMPLE   HAKMONIG   MOTION 


233 


(2)  When  the  phase-difference  is  90°,  or  one  quarter  of  a 
period. 

In  the  figure  of  §  205  let  PM  be  drawn  perpendicular  to  the 
diameter  BOB'.  The  motion  of  M  in  BOB'  is  precisely  the  same 
as  that  of  N  in  AOA',  but  M  starts  from  0  when  N  is  already  at 
A,  i.e.,  the  motion  of  if  is  a  quarter  of  a  period,  or  90°  in  phase, 
behind  that  of  N.  The  motion  of  P  is  evidently  the  resultant  of 
the  motions  of  N  and  M.  Hence : 

Two  S.H.M.'S  of  equal  amplitude  and  period,  but  differing  in 
phase  by  a  quarter  of  a  period,  or  90°,  compound  into  uniform 
motion  in  a  circle. 

If  M  had  been  a  quarter  of  a  period  in  front  of  N,  the  motion 
in  the  circle  would  have  been  in  the  opposite  direction. 

(3)  It  is  sometimes  useful  to  resolve  a  s.H.M.  into  two  uniform 
circular  motions  in  opposite  directions. 


Fig.  117. 

Let  two  points  P,  P'  start  from  A  and  describe  a  circle 
uniformly  in  opposite  directions.  The  two  circular  motions  are 
together  equivalent  to  two  S.H.M.'S  along  AOA',  and  two  others  at 
right  angles  to  AOA.  The  latter,  differing  in  phase  by  half  a 
period,  destroy  each  other.  The  resultant  is  therefore  a  S.H.M. 
along  AOA'  with  amplitude  double  the  radius  of  the  circle. 

Conversely  a  S.H.M.  may  be  resolved  into  two  uniform  circular 
motions  in  opposite  directions,  the  radius  of  the  circle  being  half 
the  amplitude  of  the  vibration. 


234  MECHANICS  [CHAP. 

This  may  be  seen  directly  from  the  figure,  for  the  resultant  of 
the  displacements  OP,  OP'  is  OR,  equal  to  twice  ON.  Such  a 
motion  is  realized  by  the  arrangement  described  in  Example  5, 
below. 

Applications  of  these  results  are  found  in  the  theory  of 
polarized  light. 

215.  Whenever  the  force  acting  on  any  mass  is  proportional 
Fundamental  Pro-  to  the  distance  from  a  fixed  point,  and  directed 
Harmo^ic^bra-6  towards  it,  so  also  will  the  acceleration  be,  and 
tion-  the  motion  will  consequently  be  Simple  Harmonic 

Motion.  Suppose  we  know,  from  a  consideration  of  the  forces 
acting,  that  in  some  given  case  the  ratio  of  the  acceleration  to 
the  distance  ON  is  //, ;  so  that  for  this  case 

a  =  -fi.ON. 

We  have  proved  (§  208)  that  if  T  is  the  periodic  time  of  the 
vibration, 

4-7T2 
•47T2 

Hence  w  =  /*> 

and  T=-^L. 

This  does  not  depend  on  the  amplitude  or  distance  of  displacement, 

,    ,      7          .7         ..    acceleration 

out  only  on  the  ratio  — r. , 

distance 

Such  vibrations,  whose  period,  or  time  of  swing,  does  not 
depend  on  the  amplitude,  are  called  isochronous,  i.e.,  executed  in 
the  same  time. 

Experiment.  Verify  this  property  with  the  apparatus  of 
experiments  of  1  and  3  of  §  210. 


XXII]  EXAMPLES  235 


EXAMPLES. 

1.  In  a  S.H.M.,  find  the  time  of  an  oscillation 

(1)  when  the  acceleration  at  a  distance  of  3  inches  from  the  centre 

is  4  ft.-sec.  units;  y 

(2)  when  the  acceleration  at  a  distance  of  25  centimeters  from  the 
centre  is  625  cm. -sec.  units; 

(3)  when  the  force  acting  on  the  body  at  a  distance  of  2  feet  is  equal 
to  its  weight. 

2.  Find  the  velocity  at  the  centre  in  each  of  the  above  cases  when  the    / 
amplitudes  are  respectively  (1)  1  foot,  (2)  10  cm.,  (3)  2  feet. 

2?r 

3.  Shew  that  in  a  S.H.M.  of  amplitude  a,  and  periodic  time  -p,  at  a 

V/* 

distance  x  from  the  centre  the  velocity  is  V /*  (a2  -  #2),  and  the  time  elapsed 

since  it  was  at  its  greatest  distance =-j=  cos"1  -. 

V/u,  a 

4.  In  a  S.H.M.  the  velocities  at  distances  5  and  12  feet  are  36  and  15  feet 
per  second  respectively ;  find  its  period  and  the  acceleration  at  the  greatest 
distance  from  the  centre. 

5.  One  end  of  a  rod  2  feet  long  is  pivoted  on  a  pin  projecting  from  the 
edge  of  the  rim  of  a  flywheel  4  feet  in  diameter.     The  other  end,  which  carries 
a  shelf,  is  attached  to  a  block  which  runs  up  and  down  a  vertical  slot  passing 
through  the  line  of  the  wheel's  axis.     Shew  that  the  motion  of  the  shelf  is  a 
S.H.M.     What  is  the  greatest  number  of  revolutions  per  minute  the  wheel  can 
make  without  causing  objects  placed  on  the  shelf  to  part  company  with  it  at 
the  top? 

6.  Hooke  discovered  that  the  tension  of  an  elastic  thread  is  proportional 
to  its  extension  per  unit  length;  so  that  if  I  is  the  unstretched  length,  the 
tension  T  required  to  stretch  it  to  a  length  I'  is,  according  to  Hooke's  law, 

rp—  rn     I  —I 

Y-/V  -r, 

where  TQ  is  a  constant  which  is  evidently  the  tension  required  to  stretch  the 
thread  to  double  its  length. 


236  MECHANICS  [CHAP,  xxn 

A  thread  30  cms.  long,  which  when  supporting  5  gms.  extends  to  a  length 
of  50  cms.,  is  fastened  30  cms.  below  a  small  hole  in  a  smooth  table.  The 
other  end  is  drawn  up  through  the  hole  and  attached  to  a  gramme  weight. 
The  weight  is  placed  on  the  table  at  a  distance  of  10  cms.  from  the  hole,  and 
let  go.  Find  the  time  of  an  oscillation,  and  the  velocity  of  the  weight  as  it 
passes  over  the  hole. 

7.  A  brass  clamp  weighing  10  gms.  is  fastened  to  one  prong  of  a  fixed 
vertical  tuning  fork,  and  when  the  fork  is  excited  by  a  violin  bow,  the  clamp 
vibrates  256  times  per  second  through  a  distance  of  1  mm.     Calculate  the 
force  (in  dynes)  exerted  on  the  clamp  at  the  extremity  of  a  vibration. 

8.  In  a  S.H.M.  of  amplitude  10  feet  and  period  15  seconds  find  the  time 
occupied  in  travelling  (1)  5  feet,  (2)  7  feet  from  a  point  of  rest. 


CHAPTER  XXIII. 


THE  SIMPLE   PENDULUM. 

216.  THE  most  familiar  case  of  a  S.H.M.  is  the  Simple  Pendulum. 
This  consists  of  a  bob  of  mass  ra 
hanging  from  a  fixed  point  by  a 
string.  Theoretically  the  mass  of 
the  bob  should  be  collected  at  a 
point,  and  the  string  should  have 
no  weight  or  mass. 

Let  I  \JQ  the  length  of  the 
string ;  6  t^e  angle  it  makes  with 
the  vertical  at  any  instant. 

The  forces  acting  on  the  bob 
are  the  weight  W  vertically  down- 
wards, and  the  tension  of  the 
string  T. 

Resolving  along  the  tangent 
at  P  we  see  that  the  only  force 
affecting  motion  along  the  circle 
is  the  component  of  W, 

TFcos(90°-0)  =  Fsin<9, 
directed    towards    0;    since   the 
tension    T    has    no    component 
along  the  tangent.  Fig.  118. 

The  acceleration  is  therefore 

Tfsinfl 
ra 

=  —  g  sin  0. 


W=  mg 


238  MECHANICS  [CHAP. 

If  6  is  very  small,  we  may  put  the  circular  measure  for  the 
sine,  so  that 

arc  OP 
^'radius  AO 

=  -f.OP.  "* 

The  acceleration  is  thus  proportional  to  the  displacement,  since 
neither  g  nor  /  depends  on  the  displacement. 
But  this  is  the  law  for  S.H.M., 

«  — 

Therefore  the  pendulum  describes  simple  harmonic  vibrations 
of  a  period  T,  such  that 

47T2  _g_ 
T2  ~  I' 


and 

9 

Observe,  this  is  only  true  for  small  angular  vibrations  of  a  few 
degrees  on  each  side  of  the  vertical.  For  small  angles  the  swings 
of  a  pendulum  are  isochronous. 

217.  Experiment.  Suspend  two  leaden  or  iron  balls  by 
threads  of  equal  length  from  a  lofty  ceiling  or  gallery.  Draw  one 
of  them  aside  a  few  inches,  and  with  the  other  hand  draw  the 
other  as  far  behind  you  as  you  can  reach.  Release  them  exactly 
at  the  same  instant,  by  opening  the  thumbs  and  fingers  so  as 
not  to  give  either  of  them  the  slightest  push.  Watch  the  first 
pendulum,  and  at  the  moment  when  it  returns  to  your  hand  close 
the  other  hand  without  looking  round.  It  will  grasp  the  other 
pendulum. 

This  principle  is  said  to  have  been  first  detected  by  Galileo 
who  timed  the  decreasing  swings  of  a  bronze  lamp  (a  masterpiece 
of  Benvenuto  Cellini)  during  a  service  in  the  cathedral  of  Pisa  by 
counting  the  beats  of  his  pulse. 

Huyghens,  who  rendered  great  services  in  the  invention  and 
perfection  of  the  pendulum  clock,  discovered  the  cycloid  and  how 
to  make  a  pendulum  swing  in  it.  In  this  curve  the  vibrations  are 


XXIII]  THE   SIMPLE   PENDULUM  239 

strictly  isochronous  whatever  their  amplitude.  But  in  practice  it 
is  better  to  use  the  simple  circular  pendulum  and  keep  the  angular 
vibrations  small. 

218.     For  a  seconds  pendulum  the  time  of  a  single  swing,  i.e., 
a  half  period,  must  be  one  second.     Hence 

Length  of  the 

Seconds 

Pendulum. 


and  Z=. 

7T* 

At  Greenwich  g  =  321 91 2  and  consequently  I  =  39'139cS3  inches. 

219.     Let  n  be  the  number  of  vibrations  with  length  l\  ri  the 
Number  of  vibra-      number  lost  for  a  length  I  +  \. 
tions  lost  in  a  day  Then   since   the   number  of  vibrations   in  a 

for  a  small  increase 

in  the  length.  given  time  vanes  inversely  as  the  periodic  time, 


n-ri        /    I 

n      ~VZ  +  X' 


or, 


neglecting  squares  of  -j . 

.-.  n'-|.n-|.  86400, 
since  a  day  of  24  hours  contains  86400  seconds. 

220.     Let  7  be  the  increase  in  g,  n'  the  number  gained. 

Number  of  vibra-  &S  before  ; 

tions  gained  for  a  f  , 

small  increase  in  n  +  n  /  Q  -\-  <y        /,          <y\* 

the  value  of  g,  ' 


240  MECHANICS  [CHAP. 

221.     For  a  point  outside  the  earth  gravity  varies  inversely 
as  the  square  of  the  distance.     (§  238.) 
At  a  height  hr  g  is  less  by  7  where 


pendulum.  _  ^ 


g  - 


R  being  the  earth's  radius. 


_ 

""*' 


But  nf  =  ^-  .  n  =  -^  .  n. 

2  R 


„ 

—  .R. 
n 


222.     For   a   point  within   the   earth   gravity  varies   as   the 
Measurement  of       distance    ^om    the    centre    (§239).      Hence    at 

depths  by  the  depth    dt 

pendulum. 

g-y  =  R-d 
g         R 


d 


223.     The  formula  T  =  27rA  /- 

V  g 


g 

Determination  of  -,  TI,I^  'j_i  i  t>  •. 

gravity  by  the          can  be  applied  to  determine  the  value  of  gravity. 

simple  pendulum. 


It  is  far  easier  to  obtain  accurate  values  of  the  length 
of  a  pendulum,  and  its  time  of  swing,  than  to  observe  the 
distance  traversed  in  one  second  by  a  falling  body;  and  if  the 
velocity  of  the  falling  body  is  diminished  by  some  device  such 
as  the  Atwood's  Machine,  errors  are  introduced  by  the  various 


XXIIl]  THE   SIMPLE   PENDULUM  241 

frictions  which  inore  than  compensate  for  the  greater  ease  of 
observation.  Newton  employed  this  method,  and  using  as  bobs 
for  his  pendulums  equal  boxes  filled  with  different  materials, 
he  was  able  to  shew  with  great  accuracy  that  the  value  of  gravity 
was  the  same  for  all  substances  at  the  same  place. 

Experiment.  A  leaden  sphere  is  hung  by  a  long  string,  and 
set  swinging  through  a  small  angle. 

Take  the  time  of  50  or  100  swings  and  find  the  time  T  of  one 
complete  double  swing. 

Measure  the  length  of  the  string  by  a  steel  tape ;  and  add  half 
the  diameter  of  the  sphere  as  determined  by  the  calipers.  Call 
this  I. 

Then  9  =  ^-1- 

In  this  experiment  the  weight  of  the  string  has  been  neglected; 
the  mass  of  the  bob  has  been  taken  as  collected  at  the  C.G.;  and  it 
is  not  easy  to  observe  with  extreme  accuracy  either  the  distance 
from  the  point  of  suspension  to  the  C.G.,  or  the  time  of  swing. 
These  difficulties  are  overcome  by  the  use  of  Kater's  Pendulum 
(§  276). 


EXAMPLES. 


1.  What  is  the  length  of  a  pendulum  beating  half-seconds  at  Greenwich  ? 

2.  The  bob  of  a  seconds  pendulum  is  screwed  up  1/32  of  an  inch.     How 
many  seconds  will  it  gain  in  a  day  ? 

3.  A  clock  with  a  seconds  pendulum  is  found  to  be  gaining  7  seconds  per 
day.     How  much  must  the  bob  be  let  down  ? 

4.  Find   the   length  of   the  seconds  pendulum  on  the  moon,   where 
#  =  150cm./sec2. 

5.  A  pendulum  beating  seconds  at  Greenwich,  where  #=32'2,  is  taken  to 
a  place  where  it  loses  4  seconds  per  day.     Find  the  value  of  g  at  this  place. 

c.  16 


242  MECHANICS  [CHAP.  XXIII 

6.  Iron  expands  -000006G  of  its  length  for  every  degree  Fahrenheit  rise 
of  temperature.     A  clock  with  its  pendulum  bob  suspended  by  an  iron  rod 
keeps  correct  time  at  62°  F.     How  many  seconds  per  day  will  it  lose  in  a 
temperature  of  80° F.? 

7.  If  a  seconds  pendulum  be  taken  to  the  top  of  a  mountain  half  a  mile 
high,  how  many  seconds  will  it  lose  in  a  day  ? 

8.  A  seconds  pendulum  carried  from  sea-level  to  the  top  of  a  mountain 
loses  15  seconds  a  day.     Taking  the  earth's  radius  at  4000  miles,  find  the 
height  of  the  mountain. 

9.  Find  the  depth  of  a  mine  at  the  bottom  of  which  a  seconds  pendulum 
loses  9  seconds  a  day. 


CHAPTER  XXIV. 

CENTRAL  FORCES.  THE  LAW  OF  GRAVITATION. 

224.  THIS  is  the  subject   of  Newton's   immortal  work,  the 
Principia.     We  shall  here  extract  the  most  fundamental  propo- 
sitions;  but   the   student   should  on   no  account  fail  to  consult 
the  original,  if  only  to  turn  over  the  pages  and  note  the  contents. 
In   no   other  way  is   it   possible   to   gain  an  impression  of  the 
immense  range  of  applications  made  by  Newton,  and  the  almost 
superhuman    power    with    which    they  are   worked    out.      The 
propositions  are  given  according  to  the  numbering  of  the  Principia, 
with  slight  simplifications  from  the  original  form. 

225.  PROPOSITION  I.     The  equable  description  of  areas. 
When   a   body   revolves   in   an   orbit   subject   to   the   action 

of  forces  tending  to  a  fixed  point,  the  areas  swept  out  by  radii 
drawn  to  the  fixed  centre  of  force  are  in  one  fixed  plane,  and  are 
proportional  to  the  times  of  describing  them. 

Let  the  time  be  divided  into  equal  parts,  and  in  the  first 
interval  let  the  body  describe  the  straight  line  AB  with  uniform 
velocity,  being  acted  on  by  no  force.  In  the  second  interval  it 
would,  if  no  force  acted,  proceed  to  c  in  AB  produced,  describing 
Be  equal  to  AB\  so  that  the  equal  areas  ASB,  BSc,  described  by 
the  radii  AS,  BS,  cS  drawn  to  the  centre  S,  would  be  completed 
in  equal  intervals.  But  when  the  body  arrives  at  B,  let  a 
centripetal  force  tending  to  S  act  upon  it  by  a  single  instantaneous 
impulse,  and  cause  the  body  to  deviate  from  the  direction  Be  and 
to  proceed  in  the  direction  BC. 

Let  cC  be  drawn  parallel  to  BS,  meeting  BC  in  (7;  then  at  the 

16—2 


244 


MECHANICS 


[CHAP. 


end  of  the  second  interval  the  body  will  be  found  at  C  (§  153) 
in  the  same  plane  with  the  triangle  A  SB,  in  which  Be,  cG  are 
drawn.  Join  SC,  and  the  triangle  SBC,  between  the  parallels  SB, 
Cc,  will  be  equal  to  the  triangle  SBc,  and  therefore  also  to  the 
triangle  SAB. 


Fig.  119. 

In  like  manner,  if  the  centripetal  force  act  upon  the  body 
successively  at  G,  D,  E,  &c.,  causing  the  body  to  describe  in 
successive  intervals  of  time  the  straight  lines  CD,  DE}  EF,  &c., 
these  will  all  lie  in  the  same  plane ;  and  the  triangle  SCD  will 
be  equal  to  the  triangle  SBC,  and  SDE  to  SCD,  and  SEF  to 
SDE. 

Therefore  equal  areas  are  described  in  the  same  plane  in 
equal  intervals ;  and  the  sums  of  any  number  of  areas  SADS, 
SAFS,  are  to  each  other  as  the  times  of  describing  them. 

Let  now  the  number  of  these  triangles  be  increased  and  their 
breadth  diminished  indefinitely ;  then  their  perimeter  ADF  will 
be  ultimately  a  curve  line;  and  the  instantaneous  forces  will 
become  ultimately  a  centripetal  force,  by  the  action  of  which  the 
body  is  continuously  deflected  from  the  tangent  to  this  curve,  and 
which  will  act  continuously ;  and  the  areas  SADS,  SAFS,  being 
always  proportional  to  the  times  of  describing  them,  will  be  so  in 
this  case.  Q.E.D. 


XXIV]  EQUABLE   DESCRIPTION    OF   AKEAS  245 

The  converse  is  given  by  Newton  as  his  second  proposition. 
It  is  easily  seen  that  if  the  body  moves  so  that  the  triangle  SBC 
is  equal  to  SBc,  then  cC  must  be  parallel  to  BS,  and  therefore  BS 
was  the  direction  of  the  impulse  at  B.  The  property  is  then 
extended  to  the  limiting  case  as  before. 

226.  Kepler's  Second  Law,  discovered  from  the  motions  of 

The  Case  of  the  Mars,   IS  '— 

Planets-  The  areas  swept  out  by  radii  drawn  from  the 

planet  to  the  sun's  centre  are,  in  the  same  orbit,  proportional  to 
the  times  of  describing  them. 

Hence  the  planets  move  as  if  acted  on  by  forces  always 
directed  to  the  sun's  centre. 

227.  Corollary.     Double  the  area  swept  out  in  the  unit  of 
time,  one  second,  in  any  orbit  is  usually  denoted  by  the  letter  h. 

Since  h  is  the  same  whether  the  body  proceeds  along  the 
tangent  with  uniform  velocity,  when  no  force  acts  on  it,  or  whether 
it  is  deflected  round  the  curve,  by  a  central  force,  it  can  be  found 
if  the  velocity.'?;  is  known.  For  let  AB  be  the  space  described  in 
one  second,  i.e.,  v  feet.  Then  h  =  2  x  area  SAB  =  v  x  p}  where  p 
is  the  perpendicular  from  S  on  the  tangent  at  A. 

Thus  ««-, 

P 

and  the  velocity  at  any  point  is  inversely  proportional  to   the 
perpendicular  from  the  centre  of  force  on  the  tangent. 

Given  the  velocity  V,  distance  R,  and  direction  of  projection  «, 
h  can  be  found. 

For      h=VRsmoL. 

The  time  occupied  in  de- 
scribing any  part  of  the  orbit 
is  then  obtained  by  dividing 
twice  the  area  swept  out  by  h. 

Conversely,  the  position  in 
the  orbit  after  a  time  t  has  S  R 

elapsed,  is  known  from  the  area  Fig.  120. 

^ht  swept  out  in  the  interval. 


246 


MECHANICS 


[CHAP. 


228.     PROPOSITION  VI.     The  Law  of  Variation  of  the  Force. 

Let  the  body  move  from   P  to   Q  in  a   very  small  interval 
of  time  t     Draw  QR  parallel      Y 
and   QT  perpendicular  to   SP. 
Then   (cf.   §  111)   if    a   is   the 
acceleration  towards  S  due  to 
the    force,    supposed    constant 
during  the  short  interval,  the 
distance     QR    fallen     through 
towards  the  centre  in  the  time 
t  is 

at2 


Fig.  121. 


But 


.      2area£PQ 
~~ 


_SP_.QT 

h      ' 

_2QR_     QR     h2 
~V~        QT~2'SP~2 

in  the  limit  when  PQ  is  indefinitely  small. 

D  7? 
The  limiting  value  of  ^-^  depends  on  the  shape  of  the  curve, 

and  the  problem  now  is  to  find  it  for  any  desired  case.  Newton 
in  his  tenth  and  eleventh  propositions  finds  it  for  (1)  an  ellipse 
described  under  a  force  tending  to  the  centre,  and  (2)  an  ellipse 
described  under  a  force  tending  to  one  focus. 

229.     PROPOSITION  X.     Let  a  body  describe  an  ellipse ;    to 
find  the  law  of  force  tending  to  the  centre. 

The  acceleration  =  ^rp^  •  7^™     (Prop.  VI). 

Draw  the  conjugate  diameter  CD,  and  the  perpendicular  PF. 
By  similar  triangles  QTu,  PFC} 

QT2     PF2 


ELLIPSE   ABOUT   ITS   CENTKE 


xxiv] 

By  the  properties  of  the  ellipse 
Qu* 


247 


PF*.CD*     A&.BC* 


Pu  .  uG          CP4 


CP4 


Fig.  122. 

But  Pu  =  QR,  and  ultimately,  when  Q  is  taken  very  close  to 

Py  uG  =  20P. 

AC*. BO*    u. 

-  Ornately. 


.*.   the  acceleration  =  limit  of  TT^-  TTTJ^ 

7i2 

.CP. 


Denoting  the  semi-axes  by  a,  6,  we  have 

A2 
acceleration  =  —^ .  CP. 

The  acceleration,  and  therefore  the  force  producing  it,  is  thus 

/?2 
proportional  to  the  distance  CP,  since  -^-2  is  a  constant  factor. 


248  MECHANICS  [CHAP. 

230.  Corollary  1.  Suppose  the  force  acting  on  a  body  is 
such  as  to  produce  an  acceleration  p  times  the  distance  of  the 
body  from  a  fixed  point,  and  that  the  body  has  been  projected 
with  an  initial  velocity  not  directed  towards  the  point.  Then  the 
body  will  describe  an  ellipse  about  that  point  as  centre;  and  since 
acceleration  =  /z,  .  CP, 


The  time  of  describing  the  complete  ellipse 

_  2  area  of  ellipse  _   Zirab   _  2-jr 
h  \ffju.ab     */fj,  ' 

The  periodic  time  is  thus  the  same  for  all  ellipses  described 
about  this  centre  of  force,  whatever  their  dimensions.  The 
dimensions  will  depend  on  the  distance  and  velocity  of  projection, 
but  will  not  affect  the  periodic  time. 

Let  different  ellipses  be  described  with  their  axes  major  along 
the  same  line;  and  let  the  bodies  be  given  smaller  and  smaller 
velocities  at  right  angles  to  this  line.  The  ellipses  will  degenerate 
into  lines  along  the  axis,  and  the  motions  become  vibrations  about 
the  centre  with  different  amplitudes  but  the  same  periodic  time. 
In  fact  we  have  arrived  by  a  different  path  at  the  case  of  Simple 
Harmonic  Motion,  with  its  fundamental  property  that  the  period 
is  independent  of  the  amplitude  (§  215). 

231.     Corollary  2.     The  velocity  v  at  any  point 
h     h.CD      h 

=  -  =  -  ^j\  =  T» 

p     p.  CD     ab 


and  the  velocity  is  proportional  to  the  semi-conjugate  diameter. 

232.  PROPOSITION  XL  Let  a  body  revolve  in  an  ellipse; 
to  find  the  law  of  force  tending  to  a  focus  of  the  ellipse. 

Let  the  force  tend  to  the  focus  8  ;  draw  QR  parallel  and  QT 
perpendicular  to  8P,  and  let  Qxv,  parallel  to  the  tangent,  cut  SP 
in  x  and  CP  in  v. 


XXIV]  ELLIPSE    ABOUT   A   FOCUS 

Draw  HI  parallel  to  CD. 
Then  SE  =  El,  since  SC=CH-, 

and 


249 


Fig.  123. 

By    Prop.    VI.    the    acceleration  =  the    limiting    value    of 
opi*  777*2  wnen  PQ  is  indefinitely  diminished. 
By  similar  triangles,  QTx,  PFE, 

QT*_PF*_PF*_BC* 

PE*~AC*~CD*    


.(1) 


since  PF.CD  =  AC.BG. 

By  a  property  of  the  ellipse 

Qv*     =CD* 
Pv.vG~  CP*' 
and  by  similar  triangles 

Pv_  =  P^_CP 
QR~  Px~  PE' 


250  MECHANICS  [CHAP. 

and  ultimately  vG  =  2CP;  Qx  =  Qv. 


CD* 


Therefore  multiplying  (1)  and  (2), 

20P2  QR  =  CP2 

QT2     2BC* 
and  -fyn  - 

2A2 
.'.   the  acceleration,  which  = 


The   acceleration,   and   therefore   the   force   producing   it,   is 
inversely  proportional  to  the  square  of  the  distance  SP  from  the 

h^a 
focus,  since  -r-  is  a  constant  factor. 


233.  Kepler's  First  Law  is  ;— 

The  planets  move  in  ellipses  having  the  sun  in  one  focus. 

The  force  exerted  upon  them  by  the  sun  must  therefore  be 
inversely  proportional  to  the  square  of  the  distance. 

This  is  the  famous  proposition  which  Hooke  guessed  at,  but 
could  not  prove  (§  94). 

234.  Corollary  1.     Suppose  the  force  acting  on  a  body  P  is 
such  as  to  produce  an  acceleration  -^^  towards  a  point  $.     Then 
the  body  will  describe  an  ellipse  about  S  as  a  focus,  such  that 


XXIV]  ELLIPSE  ABOUT  A   FOCUS  251 

The  time  T  of  describing  a  complete  ellipse 
2  x  area  of  the  ellipse 


Thus  T2  oc  a3. 

Kepler's  Third  Law  is ; — 

The  squares  of  the  periodic  times  are  proportional  to  the  cubes 
of  the  major  axes. 

It  is  to  be  inferred  that  the  accelerations  of  all  the  different 
planets  are  such  as  would  be  experienced  by  any  one  of  them  at 
the  same  distance.  In  other  words  /*  is  the  same  for  them  all. 
If  the  earth  were  substituted  for  Jupiter,  it  would  describe 
Jupiter's  orbit. 

Hence  it  is  the  same  gravity  which  acts  on  all  of  them,  a  force 

proportional   to  their  masses  (since  acceleration  =  -^  whatever 

V  MJ 

may  be  the  nature  of  their  materials,  and  inversely  proportional 
to  the  square  of  the  distance  from  the  sun. 

235.     Corollary  2.     The  velocity  v  is  given  by 

h 

"8Y' 

Draw  HZ  the  perpendicular  from  the  other  focus  on  the  tangent. 
Then  by  the  properties  of  the  ellipse 

SY_H^_     /SY^HZ  EC 

SP~ HP~  V  SP. 

~SP 


,   SP 
and 


HP 


252  MECHANICS  [CHAP. 

Denote  the  distance  SP  by  r. 
Then  tf  = 


ar 


When  the  velocity  of  projection  F,  and  the  initial  distance 
R  from  the  centre  of  force,  are  known,  the  axis  major  is  determined 
from 


2 

If  F2  =  IJL  .  -£  ,  a  is  infinite,  and  the  ellipse  becomes  a  parabola. 

2 

If  F2  >  //,  .  -~  ,  a  is  negative,  and  the  orbit  is  a  hyperbola. 

Newton  points  out  these  cases,  but  for  the  sake  of  their  im- 
portance, repeats  the  proof  of  Prop.  XI.  in  a  form  adapted  to 
parabolic  and  hyperbolic  orbits  ;  and  later  on  he  shews  that  the 
orbits  of  comets  are  either  hyperbolas  (in  .which  case  they  only 
once  visit  the  solar  system)  or  such  extended  ellipses  that  they 
approximate  to  parabolas,  so  that  the  comets  only  return  at  very 
long  intervals. 

236.  The  motions  of  all  the  heavenly  bodies  were  thus 
accounted  for  on  the  hypothesis  that  they  were  acted  on  by 
central  forces  varying  inversely  as  the  squares  of  the  distances 
and  directed  towards  the  sun. 

Before  this  hypothesis  could  be  generalized  into  the  law  of 
universal  gravitation,  Newton  had  to  calculate  how  a  sphere 
composed  of  particles  each  attracting  every  other  particle  with  a 
force  proportional  to  the  product  of  the  masses  and  inversely  pro- 
portional to  the  squares  of  the  distances  would  behave  to  another 
sphere  similarly  composed.  This  was  effected  in  the  beautiful 
series  of  propositions  forming  Section  xn.  of  the  Principia.  The 
most  important  are  here  given  with  slight  changes  in  accordance 
with  modern  methods.  Newton  had  hardly  hoped  for  such 
a  simple  result  but  had  expected  to  treat  the  heavenly  bodies  as 
particles  only  as  an  approximation  to  the  truth,  permissible  on 
account  of  their  great  distances. 


XXIV] 


ATTRACTION   OF   A    SPHERE 


253 


Q 


Fig.   124. 


237.  Attraction  of  a  thin  spherical  shell  on  a  particle  inside 
it.     (Principia,  Sec.  xn.  Prop.  70.) 

Let  the  particle  be  at  0. 

Consider  a  cone  of  very  small 
angle  with  0  as  vertex,  cutting 
the  surface  at  P  and  Q. 

The  surface  is  equally  inclined 
to  the  chord  PQ  at  P  and  Q,  so 
that  the  areas  cut  out  by  the  cone 
at  P  and  Q  are  proportional  to  the 
squares  of  their  distances  from 
the  vertex,  i.e.  as  OP2  is  to  OQ2. 

But  the  attractions  of  particles 
at  P  and  Q  are  inversely  as  OP2 
and  OQ2.     Therefore  the  attrac- 
tions of  the  areas  at  P  and  Q  are  equal,  for  they  are  directly  as 
the  areas  and  inversely  as  the  squares  of  the  distances. 

These  attractions  mutually  destroy  each  other ;  and  the  same 
is  true  for  every  cone  drawn  through  0. 

Thus  the  attraction  of  the  whole  shell  on  0  is  zero. 

And  since  a  shell  of  definite  thickness  may  be  conceived  as 
made  up  of  a  number  of  concentric  thin  shells,  the  same  is  true 
for  the  attraction  of  a  thick  shell  upon  a  point  inside  its  inner 
boundary. 

238.  Attraction  of  a  thin  spherical  shell  on  a  particle  outside 
it.     (Principia,  Sec.  xn.  Prop.  71.) 

Let  P  be  the  position  of  the  particle,  and  0  the  centre  of  the 
sphere. 

From  symmetry  the  resultant  attraction  must  be  along  PO. 

Let  p  be  the  mass  of  the  shell  per  unit  area,  r  the  radius  of  the 
sphere. 

Join  OP  and  divide  it  at  B  so  that 

OB  :  r  ::  r  :  OP. 

Consider  a  cone  through  B  cutting  out  a  small  area  S  at  Q. 
Its  attraction  is 

along  PQ. 


254 


MECHANICS 


[CHAP. 


The  component  of  this  along  PO  is 
v_PScosOPQ 

~PQr~ 
But         =  -      ;  so  that  OBQ,  OQP  are  similar  triangles,  and 


PQ~OP~  OP' 

pr*    8  cos  OQB 


OQB  =  z  OPQ. 

Also 


Thus 


Fig.  125. 


Let  a  sphere  be  drawn  round  B  as  centre,  with  radius  BR,  and 
another  with  any  fixed  radius  a. 

The  cone  will  cut  from  these  spheres  the  areas  RT,  rt. 
But  area  RT  =  8  cos  QRT=  8  cos  OQB. 

Y-  pr*      area  RT 
~"OP~2X~'B^~ 

pr2      area  rt 
~  OP2  X  ~tf~ ' 

Let  the  same  be  done  for  every  small  area  of  the  shell.     Then 
the  total  resultant  attraction  along  PO 


x  — 


whole  surface  of  sphere,  radius  a 


OP2 

pr2       47raa 
OP'2  X    a8 


xxiv] 


ATTRACTION    OF   A   SPHERE 


255 


But  this  is  the  attraction  of  the  mass  of  the  shell  4<7rr2p 
collected  at  the  centre  0. 

Hence  the  attraction  of  the  shell  is  the  same  as  if  it  were  all 
collected  at  its  centre. 

Since  a  solid  sphere  may  be  conceived  as  made  up  of  concentric 
shells,  the  same  is  true  for  a  solid  sphere,  whether  its  density  is 
the  same  throughout,  or  depends  on  the  distance  from  the  centre. 

239.     Corollary.     (Principia,  Prop.  73.) 

The  attraction  of  a  homogeneous  solid  sphere  upon  a  particle 
inside  it  is  proportional  to  the  dis- 
tance from  the  centre. 

For  draw  a  concentric  sphere 
through  the  point  P  distant  r 
from  the  centre.  Then  the  shell 
outside  this  sphere  has  no  attraction 
on  a  particle  at  P  (Prop.  70).  The 
attraction  of  the  rest  is 


(Prop.  71) 


Fig.  126. 


EXAMPLES. 

1.  If  a  straight  tunnel  could  be  driven  through  the  centre  of  the  earth, 
shew  that  a  cannon  ball  dropped  into  it  would  reach  the  antipodes  in  about 
42|  minutes.      Take  #  =  32-2;  earth's  radius  =  4000  miles;  and  neglect  the 
resistance  of  the  atmosphere. 

2.  Shew  that  the  time  would  be  the  same  for  a  train  of  cars  running  on 
smooth  rails  through  a  straight  tunnel  to  any  part  of  the  earth's  surface. 


3.     Shew  that  if  the  earth's  velocity  in  her  orbit  were  increased  by  about 
one  half,  she  would  describe  a  parabola  about  the  sun. 


256  MECHANICS  [CHAP,  xxiv 

4.  Shew  that  if  a  body  were  projected  from  the  earth  with  a  greater 
velocity  than  about  7  miles  per  second,  it  would  not  return  to  her. 

5.  The  sy  nodical  period  of  a  planet  exterior  to  the  earth,  i.e.  the  interval 
between  two  successive  conjunctions,  or  moments  when  they  are'  in  the  same 
direction  from  the  sun,  is  S.    Shew  that  if  ^and  P  be  the  times  of  revolution 
of  the  earth  and  the  planet  about  the  sun, 


Hence  find  the  time  of  revolution  of  Jupiter,  whose  synodical  period  is 
observed  to  be  398'88  days,  the  time  for  the  earth  being  365-25  days  ;  and 
from  Kepler's  Third  Law  deduce  the  mean  distance  of  Jupiter  from  the  sun, 
that  of  the  earth  being  92,390,000  miles. 


CHAPTER  XXV. 

IMPACT  AND   IMPULSIVE  FORCES. 

240.  ACCORDING  to  the  Second  Law  of  Motion  the  effect  of 
a  force  -is  to  determine  at  every  instant  a  rate  of  change  of  velocity 
in   a   body   on   which   it   acts,   i.e.   an    acceleration.     From    the 
acceleration  we  can  find  by  the  kinematic  formulae,  the  velocity 
of  the  body  at  any  subsequent  time,  and  the  distance  it  will  have 
moved  in  the  interval ;  and  thus  the  whole  effect  of  the  force  is 
known. 

There  are  cases,  as  when  two  billiard  balls  collide,  or  a  ball 
receives  a  blow  from  a  cricket  bat,  where  the  whole  time  of  action 
is  so  excessively  short  that  we  can  no  longer  follow  the  process  in 
detail;  yet  during  the  momentary  contact  forces  are  called  into 
play,  rising  from  zero  to  a  high  value,  and  dying  away  to  zero 
again  according  to  unknown  laws,  so  that  a  great  and  apparently 
instantaneous  change  of  velocity  takes  place.  Such  forces  are 
called  impulsive,  and  these  cases  of  impact  or  collision  require 
a  somewhat  different  treatment. 

The  very  fact  that  prevents  us  from  applying  the  previous 
method,  the  shortness  of  the  time,  relieves  us  of  half  the  difficulty. 
It  is  not  necessary  to  calculate  where  the  body  will  be,  since  it 
has  not  time  to  change  its  place  appreciably  during  the  blow.  We 
know  where  it  is,  and  it  suffices  to  find  the  total  change  of 
velocity.  The  subsequent  motion  under  finite  forces  can  then  be 
calculated  as  before. 

241.  To  fix  our  ideas,  let  us  suppose  that  a  sphere  A  of  mass 
M,  moving  to  the  right  with  velocity  U  overtakes  and  impinges 
directly  upon  another  sphere  B  of  mass  m,  also  moving  to  the 

c.  17 


258  MECHANICS  [CHAP. 

right  with  smaller  velocity  u.    By  direct  impact,  is  meant  that  the 
line  of  centres  is  the  line  of  motion  for  both  balls. 


Fig.  127. 

The  blow  will  change  both  velocities.  Let  the  new  velocities, 
immediately  after  impact,  be  V  and  v. 

However  irregularly  the  pressure  between  the  balls  may  vary 
during  the  impact,  at  every  instant  the  action  of  the  first  ball  upon 
the  second  will,  by  the  Third  Law  of  Motion,  be  met  by  an  equal 
and  opposite  reaction  of  the  second  upon  the  first.  And  therefore 
the  total  impulse  for  the  whole  time  of  contact  will  be  the  same 
for  the  reaction  as  for  the  action,  but  in  the  opposite  direction. 

By  the   Second  Law  change  of  momentum  is  equal  to  the 
impulse ;  so  that  the  momentum  of  B  will  be  increased  precisely 
as  much  as  the  momentum  of  A  is  diminished,  and  the  sum  of  the 
momenta  after  impact  is  the  same  as  it  was  before.     Thus 
M  V  +  mv  =  M  U  +  mu. 

Or  instead  of  considering  the  effect  on  each  ball  separately,  let 
the  system  considered  be  the  two  balls  taken  together.  During 
the  infinitely  short  time  of  contact  any  finite  external  forces  that 
may  be  acting,  such  as  gravity,  have  no  time  to  produce  a  change 
of  momentum.  The  momentum  of  the  system  is  therefore 
unaltered,  and  again 

MV  +  mv  =  MU+mu (1). 

A  second  relation  is  required  to  determine  the  two  unknown 
quantities  V  and  v.  For  this  we  recur  to  experiment. 

242.  If  the  balls  are  made  of  clay,  putty,  or  similar  substances, 
i.  inelastic  tnev  are  squeezed  out  of  shape  by  the  blow,  but 
Bodies.  shew  no  tendency  to  recover  their  form  and  to 


XXV] 


IMPACT  AND   IMPULSIVE   FORCES 


259 


thrust  each  other  apart.     They  adhere  together  and  move  forward 
as  one  mass. 
In  this  case 

V  =  v  ..............................  (2). 

Solving  (1)  and  (2)  we  have 

T7,  M  U  +  mu 

I      —    »,  _ 

V     •—   (j  —         ~   -  % 

M  +m 

The  total  impulse  R  between  the  balls  is  equal  to  the  change 
of  momentum  of  either  ball.     Thus 

R  =  m(v-u)=-M(V-U) 


243. 


M 


m 


II.     Elastic 
Bodies. 


Balls  of  ivory,  steel,  glass,  and  most  other  substances 
instantly  recover  their  form  and  thrust  each  other 
apart.     Newton  discovered  the  second  relation  for 
these  elastic  substances  by  means  of  an  experiment  remarkable 
for  its  simplicity  and  elegance. 
Experiment. 


Fig.   128. 


17—2 


260 


MECHANICS 


[CHAP. 


Let  two  balls  be  suspended  from  a  lofty  support  by  long 
threads  in  such  a  manner  that  they  rest  in  contact  with  their 
centres  in  the  same  horizontal  line  when  the  threads  are  precisely 
equal  and  parallel.  To  ensure  that  they  shall  move  in  the  same 
plane  it  is  best  to  use  a  V-shaped  suspension  instead  of  a  single 
thread  for  each  ball. 

Draw  the  balls  aside  through  different  arcs,  and  release  them 
simultaneously  by  gently  opening  the  fingers.  They  will  meet  at 
the  lowest  point.  For  by  the  property  of  the  pendulum  (§  216) 
the  time  of  describing  any  arc,  small  compared  with  the  length  of 
the  suspension,  is  independent  of  the  length  of  the  arc. 

The  velocities  with  which   they  meet   are  easily  calculated. 
For  they  are  those  due  to  the 
vertical  falls.     Thus  M.'  ........ 


But  PM  .  MM'  =  M.  A\  and 
since  AP  is  small  compared 
with  the  radius  OA  (  =  a), 

MA2 

-^:  —  very  approximately. 


A 
B 

Fig.  129. 


..      =      . 
2a 

and  v  is  proportional  to  AM. 

Set  a  metre  scale  hori- 
zontally  just  beneath  the  two 
balls,  so  that  when  they  are 
drawn  aside  to  P  and  Q,  the 

distances  AM,  BN  can  be  read  off;  let  them  be  H,  h.     These  will 
serve  as  measures  of  the  velocities  at  impact. 

Similarly,  let  the  horizontal  distances  K,  k,  to  which  the  balls 
rebound,  be  noted.  These  measure  on  the  same  scale,  the 
velocities  of  rebound  at  A,  By  since  a  ball  will  rise  on  a  curve 
to  that  height  from  which  it  must  fall  to  gain  its  velocity  of 
projection. 

With  a  little  practice  two  observers,  one  to  release  the  balls 
simultaneously  from  measured  distances  AM,  BN,  and  observe 
the  distance  K  of  rebound,  while  the  other  observes  the  rebound  k, 


XXV]  IMPACT   AND   IMPULSIVE    FORCES  261 

can  obtain  very  consistent  results  for  a  series  of  experiments,  and 
still  greater  accuracy  may  be  attained  if  the  balls  are  held  in 
position  by  a  very  fine  thread,  and  released  from  rest  by  burning 
it,  instead  of  releasing  them  by  hand. 

Let  H,  h,  K,  k  be  given  the  signs  of  the  velocities  they 
measure.  Thus  H  and  k  representing  velocities  to  the  right 
will  be  positive,  and  h  and  K  negative.  Then  it  will  be  found 
that 


or,  numerically,  (K  +  k)  =  e(H  +  h),  where  e  is  a  constant  fraction 
for  all  values  of  H  and  K,  so  long  as  balls  of  the  same  material 
are  used. 

Translating  this  into  velocities,  we  have 

(V-v)  =  -e(U-u\ 

i.e.,  the  relative  velocity  after  impact  is  a  fixed  fraction  of  the 
relative  velocity  before  impact,  and  is  reversed  in  direction. 

If  balls  of  other  materials  are  used,  the  value  of  the  fraction  e 
will  be  different,  but  the  same  law  will  hold  good,  e  is  called  the 
coefficient  of  restitution  for  the  given  materials. 

This  is  the  relation  discovered  by  Newton.  The  beautiful 
ingenuity  with  which  the  property  of  the  pendulum,  and  Galileo's 
theory  of  motion  on  a  smooth  curve  are  employed,  will  be  best 
appreciated  by  the  student  if  he  will  try  to  think  how  else  he 
could  project  two  balls  so  as  to  be  sure  they  will  meet  at  an 
expected  point  where  they  may  be  observed  ;  control  and  vary 
the  velocities  of  impact  ;  and  measure  the  instantaneous  velocities 
of  rebound. 

244.  When  the  coefficient  e  has  been  experimentally  deter- 
mined for  balls  of  given  materials,  the  problem  of  collision  is 
easily  solved. 

By  the  dynamical  principle  of  momenta  we  have 

MV+mv  =  MU  +  mu  ...................  ..(1). 

By  Newton's  experimental  result 

V-v  =  -e(U-u)  ........................  (2). 


262  MECHANICS  [CHAP. 

MU+mu  —  em  (U  —  u) 


Tr 
Whence  F= 


,, 

M  +m 


__ 


For  inelastic  bodies  e  =  0,  and  as  in  §  242 

MU  +  mu 
V  =  v  =  — ^ —     — . 
M  +  m 

For  balls  of  glass        e  =  0'94 
„       „      .,  ivory        e  =  0'81. 
„       „      „  cast  iron  e  =  0'66. 
,  lead          e  =  0-2. 


245.  The  physical  meaning  of  the  coefficient  e  may  be  seen 
as  follows. 

Before  impact  the  centres  of  gravity  of  the  balls  are  ap- 
proaching, and  after  it  they  are  separating.  There  must  have 
been  some  moment  during  impact  at  which  they  were  relatively 
at  rest,  and  the  balls  on  the  whole  had  the  same  velocity.  Let  us 
call  this  the  moment  of  greatest  compression.  Let  R  be  the 
total  impulse  between  the  balls  up  to  that  moment,  i.e.  during 
compression  ;  and  R'  the  further  impulse  while  they  are  recovering 
their  shape,  i.e.,  during  restitution. 

At  the  moment  of  greatest  compression  both  balls  have  the 
same  velocity  V  ;  hence,  as  for  inelastic  bodies, 

(M  +  m)V'  =  MU+mu, 

M  U  •+  mu 


The  impulse  R  is  measured  by  the  change  of  momentum  of 
either  ball  up  to  the  moment  of  greatest  compression.  Taking 
the  first  ball 

R  =  MV'-MU 


Mm(U-u) 


XXV]  LOSS   OF   ENERGY   ON    IMPACT  2G3 

The  impulse  R'  is  the  further  change  of  momentum  after  the 
moment  of  greatest  compression. 

—  em(U—  u)     MU+mu 


M  +  m  M  +  m 

eMm(U-u) 

M+m 
=  -eR. 

The  fraction  e  thus  measures  the  ratio  of  the  impulse  of  the 
elastic  forces  by  which  the  balls  recover  their  form  to  the  impulse 
of  the  force  used  in  compressing  them. 

246.     The  energy  before  impact  is 

Loss  of  Energy  E  =  J  (M  U2 

during  an  impact.  The  energy  after  impact  i 


_  M  \MU+mu-em(U-u)}2     m  (MU+mu  +  eM(U-u)} 2 
=  2\  M  +  m  j~l"2J~       ~~T~ 

_  (M  +  m)  (MU  +  mu)2  +  (m  +  M)e2.  Mm  (  U  -  u)2 

=  (MU  +  mu)2  +  ez  Mm  (U-u)2 

2  (M+  m) 

If    e  =  1,  i.e.  if  the  balls  are  perfectly  elastic, 
F/  _  (M U  +  mu)2  +  Mm  (U-u)2 

2  (M+m) 
_  M'2  U2  +  m2u?  +  Mm  U2  +  Mmu2 

2  (M+m) 
MU2  +  mu2 


Thus  the  total  kinetic  energy  of  the  two  balls  is  the  same 
after  impact  as  before. 

In  practice  e  is  always  less  than  1.     Then 
E,  =  (M  U  +  mu)2  +  Mm  (  U  -  u)z  _  (1  -e8)  Mm(U-uf 

~~2(M+m) 


(M  +  m) 


264  MECHANICS  [CHAP. 

The  second  term  on  the  right-hand  side  is  necessarily  positive, 
so  that  E'  is  less  than  E. 

The  energy  of  motion  after  impact  is  thus  less  than  it  was 
before.  The  rest  has  been  transformed  into  energy  of  heat  and 
sound,  or  permanent  deformation  of  the  body  against  its  cohesive 
forces. 

247.  Whether  the  energy  so  transformed  is  to  be  considered 
as  lost  or  not,  depends  upon  the  purpose  with  which  the  blow  is 
struck.  If  it  be  desired  to  drive  a  pile  into  the  ground,  or  a  nail 
into  a  block  of  wood,  the  energy  converted  into  sound,  heat,  and 
permanent  deformation  is  wasted.  But  for  shaping  a  rivet  by 
hammering,  or  forging  a  block  under  the  stearn  -hammer,  this 
contains  the  valuable  part  of  the  energy. 

In  practice  the  object  struck  is  generally  at  rest,  and  may  be 
taken  as  inelastic.  In  this  case  w  =  0;  e  —  0'}  and  the  second 
term  in  the  expression  for  E'  becomes 

MmU* 

2  (M  +  m)  ' 

The  energy  of  the  hammer  was  —  ^  —  . 

4 

,  Energy  transformed         m 

I  h.US  -  -  rr         !  -  —    —  Trp    ;          • 

1  otal  energy  M  +  m 

For  driving  piles  or  nails  this  must  be  made  as  small  as 
possible  ;  so  that  M  should  be  great  compared  with  m  ;  i.e.,  the 
ram  of  the  pile-driver  or  the  hammer-head,  should  have  great  mass 
compared  with  the  pile  or  nail  to  be  driven. 

For  shaping  rivets  and  forgings  -^  —  -  must  be  as  large  as 

"~ 


possible;  i.e.,  m  must  be  large  compared  with  M.  This  means 
that  the  anvil  must  be  much  heavier  than  the  hammer.  The 
hammer  again  should  be  heavy  with  slow  velocity,  rather  than 
light  with  high  velocity,  in  order  to  avoid  waste  of  energy  in 
sound  and  heat,  and  convert  as  much  as  possible  into  permanent 
deformation. 

These  were  the  considerations  which  guided  Nasmych  in  the 
construction  of  his  steam  hammer.     He  says*  :  — 
*  Wright's  Mechanics. 


XXV]  IMPACT   OF    A   STREAM  265 

"  Pile  driving  had  before  been  conducted  on  the  cannon  ball 
principle.  A  small  mass  of  iron  was  drawn  slowly  up,  and 
suddenly  let  down  on  the  head  of  the  pile  at  a  high  velocity. 
This  was  destructive,  not  impulsive  action.  Sometimes  the  pile 
was  shivered  into  splinters  without  driving  it  into  the  soil  ;  in 
many  cases  the  head  of  the  pile  was  shattered  into  matches,  and 
this  in  spite  of  the  hoop  of  iron  about  it.  On  the  contrary 
I  employed  great  mass  and  moderate  velocity.  The  fall  of  the 
steam  hammer  block  was  only  3  or  4  feet,  but  it  went  on  at 
80  blows  a  minute,  and  the  soil  into  which  the  pile  was  driven 
never  had  time  to  grip  or  thrust  it  up." 

248.     In  this  case  v  and  u  are  both  zero  in  the  equation 

V-v  =  -e(U-u), 

Impact  of  a 

sphere  on  a  fixed  go  that  V  =  —  elf, 

plane. 

and  the  sphere  is  reflected  with  velocity  diminished 
in  the  ratio  e  :  1. 

The  impulsive  pressure  between  the  sphere  and  the  plane  is 
measured  by  the  change  of  momentum  produced  in  the  sphere, 
and  is  equal  to 

MV-(-eMV) 


For  example,  let  a  stream  of  water  one  inch  in  diameter  strike 
a  wall  directly  with  velocity  v  feet  per  second. 

The  volume  of  water  striking  the  wall  per  second  is  v  x  TT 
cubic  feet,  and  its  mass  at  1000  oz.  per  cubic  foot  is 


The  change  of  momentum  per  second  is 

mv  (1  +  e)  =  i$p  TT  (Jj)a  .  tf  (1  +  e), 

and  this  measures  the  steady  pressure  between  the  water  and  the 
wall. 

In  hydraulic  mining  the  impact  of  powerful  streams  of  water 
is  largely  employed  for  cutting  away  hill  sides. 

Very  important  applications  of  this  theory  are  found  in  the 
Kinetic  Theory  of  Gases,  in  which  the  known  relations  between 


266 


MECHANICS 


[CHAP. 


the  volume,  pressure,  and  temperature  of  a  gas  are  explained 
by  regarding  it  as  an  assemblage  of  innumerable  free  particles  in 
rapid  motion. 


Oblique  Impact. 


249.  Cases  of  oblique  impact  between  smooth  bodies  are 
treated  by  resolving  the  motions  along  the  line 
of  centres  and  perpendicular  to  it.  The  velocities 
along  the  line  of  centres  can  be  calculated  by  the  laws  of  direct 
impact ;  the  velocities  at  right  angles  to  the  line  of  centres  are 
unchanged  since  there  is  no  friction.  The  resultant  final  velocities 
are  then  found  by  compounding. 

The  case  of  impact  on  a  plane  will  suffice  for  illustration. 

Let  u  be  the  velocity  with 
which  a  particle  moving  along 
PO  strikes  a  fixed  plane  at  0. 
This  is  equivalent  to  u  cos  a 
perpendicular  to  the  plane,  and 
u  sin  a.  parallel  to  it. 

After  impact  the  velocities    


o 

Fig.  130. 


will  be  —  eucosa,  and  wsina 
respectively.  The  resultant  ve- 
locity will  be 

u  Ve2  cos2  a.  +  sin2  a, 

and  the  direction  of  motion  will  make  an  angle  6  with  the  normal, 
where 

u  sin  a        1 

tan  9  =  -          —  =  -  tan  a. 
e .  u  cos  a     e 


XXV]  EXAMPLES  207 


EXAMPLES. 

1.  A  mass  of  10  Ibs.  moving  5  feet  per  second  overtakes  a  mass  of  4  Ibs. 
moving  in  the  same  line  3  feet  per  second.     Find  the  velocities  after  impact, 
and  the  total  impulse  between  the  masses  (1)  when  they  are  inelastic, 
(2)  when  the  coefficient  of  restitution  is  '6. 

2.  Two  balls  (e  =  |)  of  masses  4  Ibs.  and  3  Ibs.  impinge  from  opposite 
directions  with  velocities  of  6  and  8  feet  per  second  respectively.     Find  the 
velocities  after  impact,  and  the  loss  of  kinetic  energy. 

3.  Shew  that  if  two  perfectly  elastic  balls  of  equal  mass  impinge  directly 
they  exchange  velocities. 

4.  A  perfectly  elastic  ball  is  projected  against  the  first  of  a  number  of 
exactly  similar  balls  arranged  in  a  straight  line,  each  in  contact  with  the 
next.     Shew  that  the  last  ball  will  fly  off  with  the  velocity  of  the  impinging 
ball,  the  others  remaining  at  rest. 

5.  A  train  of  cars  loaded  to  equal  weights  is  standing  at  rest  with  a 
space  of  three  inches  between  each  car  and  the  next,  the  utmost  the  couplings 
will  allow.     Another  car  of  equal  weight  is  shunted  on  to  it  behind  at  1  mile 
an  hour.     Shew  that  if  the  buffers  are  perfectly  elastic  and  the  couplings 
inelastic,  a  passenger  will  experience  two  forward  jerks  before  getting  into 
uniform  motion,  and  that,  if  each  car  is  60  feet  long,  the  mean  speed  with 
which  the  first  impulse  travels  through  the  train  is  241  miles  an  hour.     Find 
also  the  interval  between  the  two  jerks  for  a  passenger  in  the  sixth  car  from 
the  front ;  and  the  velocity  with  which  the  train  finally  starts  off  if  there  are 
20  cars  in  all. 

6.  A  fire  engine  can  project  a  stream  of  water  l£  inches  in  diameter  to 
a  height  of  144  feet.     If  the  stream  is  turned  horizontally  on  to  a  wall  find 
the  pressure  on  the  wall,  taking  e=£,  and  a  cubic  foot  of  water  to  weigh  1000 
ounces. 

7.  The  ram  of  a  pile-driver  weighs  200  Ibs.  and  drives  a  pile  f  inch 
into  the  ground  after  falling  16  feet.     What  steady  pressure  could  the  pile 
support  ? 

8.  How  many  blows  of  a  steam  hammer  weighing  400  Ibs.  (with  a  stroke 
of  2  ft.,  pressure  of  steam  80  Ibs.  per  square  inch,  and  piston  diameter 
8  inches)  would  be  required  to  drive  the  pile  in  question  7  another  6  inches 
into  the  ground  ? 


268  MECHANICS  [CHAP,  xxv 

9.  Shew  that  to  hit  a  billiard  ball  after  one  reflection  from  a  cushion  you 
should  aim  at  an  imaginary  ball  as  far  behind  the  cushion  as  the  real  one  is 
in  front  of  it,  assuming  that  the  cushions  are  perfectly  elastic. 

10.  In  the  "half-ball"  stroke  at  billiards  the  player  aims  so  that  the 
centre  of  his  ball  would  pass  through  the  extreme  edge  of  the  ball  aimed  at. 
Assuming  the  balls  perfectly  elastic,  find  the  inclinations  of  their  directions 
of  motion  after  impact  to  the  line  joining  their  centres  before  either  was 
struck,  if  the  distance  between  centres  was  5  feet  and  the  diameter  of  the 
balls  2  inches. 

11.  A  projectile  weighing  600  Ibs.  is  fired  from  a  gun  weighing  12  tons 
with  a  muzzle  velocity  of  2000  feet  per  second.     What  is  the  velocity  of  recoil 
of  the  gun  ? 

12.  If  the  earth,  when  at  the  end  of  the  minor  axis  of  her  orbit,  collided 
directly  with  a  comet  of  one-millionth  of  her  mass  and  twice  her  velocity, 
and  absorbed  it,  what  would  be  the  change  in  the  major  axis  of  her  orbit  ? 
Hence  find  the  change  in  the  length  of  the  year. 


BOOK  IV. 

THE  ELEMENTS  OF  RIGID   DYNAMICS. 


CHAPTER   XXVI. 

THE  COMPOUND   PENDULUM. 

250.  UP  to  this  point  the  masses  whose  motions  have  been 
investigated  have  been  supposed  to  be  collected  at  single  points, 
capable  only  of  a  motion  of  translation  from  place  to  place.  This 
part  of  the  subject  is  called  the  Dynamics  of  a  Particle. 

What  is  known  as  Rigid  Dynamics,  or  the  Dynamics  of  a 
Rigid  Body,  took  its  rise  out  of  the  problem  of  the  Compound 
Pendulum.  Neither  particles  (in  the  sense  of  masses  collected  at 
mathematical  points)  nor  absolutely  rigid  bodies  are  found  in 
nature.  But,  so  far  as  motion  of  translation  is  concerned,  real 
bodies  behave  as  if  their  masses  were  collected  at  their  centres  of 
gravity;  and  most  solids  are  sufficiently  rigid  to  justify  their 
treatment  as  perfectly  rigid  bodies,  at  all  events  for  a  first 
approximation.  We  can  afterwards  go  on  to  take  account  of  their 
deformations,  and  then  we  enter  on  the  Theory  of  Elasticity. 

The  Simple  Pendulum  treated  in  Chapter  xxm  consisted  of 
a  bob,  whose  mass  was  supposed  to  be  collected  at  its  centre, 
suspended  by  a  string  without  weight.  We  cannot  construct  such 
an  ideal  pendulum,  but  we  can  go  very  near  it  by  taking  a  very 
heavy  bob,  and  suspending  it  by  a  fine  but  strong  wire. 

Any  solid  object  suspended  from  a  horizontal  axis  is  found 
to  perform  oscillations  exactly  like  those  of  a  simple  pendulum. 

Drill  a  hole  through  a  flat  board,  and  pass  a  knitting  needle 
through  it.  Hang  a  bullet  just  in  front  of  the  board  by  a  thread 
wound  round  the  needle.  Hold  the  needle  horizontal,  and  set 
them  swinging  by  a  jerk  to  one  side.  If  the  time  of  swing  of 


272 


MECHANICS 


[CHAP. 


Fig.   131. 


the  bullet  is  less  or  greater  than  that  of  the  board,  unwind  or 

wind  up  the  thread  till  they  become 

equal.     Mark  the  spot  on  the  board 

exactly  behind  the  bullet  when  both 

are  at  rest.     Then  set  them  swinging 

by  another  jerk.     It  will   be   found 

that  the  bullet  remains  on  the  marked 

spot  throughout  the  motion ;  nor  can 

it  be  made  to  leave  the  spot  by  any 

amount  of  jerking,  however  irregular. 

As     dynamical     systems    oscillating 

about   this   axis   under   gravity   the 

board  and  the  suspended  bullet  are 

identical. 

Such  a  solid  object  as  the  board 
is  called  a  Compound  Pendulum.  An  ideal  pendulum  of  the  same 
time  of  swing  as  the  bullet  is  called  the  Simple  Equivalent  Pen- 
dulum for  the  board. 

251.  The  problem,  first  solved  by  Huyghens,  but  attempted 
by  most  of  the  leading  mathematicians  of  his  time,  was  to  find  by 
calculation  the  time  of  swing  of  a  compound  pendulum  of  any 
shape.  The  difficulty  to  be  overcome  was  this.  The  particles  of 
the  object  at  different  distances  from  the  axis  would,  if  free  to 
swing  separately,  perform  oscillations  in  different  times,  those  near 
the  axis  vibrating  rapidly,  and  those  far  away  more  slowly.  But 
they  are  constrained,  as  parts  of  a  rigid  body  held  firmly  together 
by  internal  forces,  all  to  vibrate  in  the  same  time.  A  compromise 
has  to  be  effected,  and  the  whole  body  swings  at  some  intermediate 
rate,  which  in  fact  may  be  determined  by  the  experiment  with  the 
bullet.  How  can  this  rate,  or  the  length  of  the  simple  equivalent 
pendulum,  be  calculated  from  the  known  dimensions  and  structure 
of  the  body,  and  the  position  of  the  axis  ? 

The  idea  by  which  Huyghens  reached  his  solution  was  this. 
Suppose  that  at  some  moment  when  the  body  is  passing  through 
its  lowest  position,  i.e.,  when  the  centre  of  gravity  is  in  the  vertical 
through  the  axis,  the  whole  body  could  be  released  from  the 
internal  forces,  and  resolved  into  its  separate  particles,  so  that 


XXVJ] 


THE   COMPOUND   PENDULUM 


273 


each  could  swing  on  its  own  account.  Then  the  height  through 
which  the  centre  of  gravity  of  the  system  of  separate  pendulums 
formed  by  the  free  particles  will  rise  in  consequence  of  their 
existing  motions  must  be  the  same  as  the  height  to  which  the 
centre  of  gravity  of  the  body  as  a  whole  actually  rises.  Huyghens 
sees  this  as  an  extension  of  Galileo's  principle  of  Work  (§§41,  66). 
There  cannot  be  a  rise  of  weights  on  the  whole,  effected  on  their 
own  account,  without  the  aid  of  external  forces.  If  the  centre  of 
gravity  rose  more,  or  less,  in  the  one  case  than  in  the  other,  it 
would  be  possible  to  make  a  perpetual  motion  and  even  produce 
work  out  of  nothing,  as  Galileo  argued  in  the  case  of  motion  on  an 
Inclined  Plane  (§  66). 


252.  For  simplicity,  consider  a  straight  rod  free  to  swing 
Problem  of  the  about  one  end  C.  Let  the  rod  swing  from  CB 
straight  Rod.  to  Q& ^  and  let  CO  =  L  be  the  length  of  the  simple 

equivalent  pendulum,  so  that  a  bullet  hung  by  a  thread  CO  would 
swing  to  CO'. 

Let  v  be  the  velocity  of  the  bullet  as  it  passes  through  0.     At 
this  moment  the  velocity  of 
any  other  point  in  the  rod, 
distant  r  from  the  axis  (7,  is 

r 
x.v. 

Suppose  that,  the  rod  is 
now  broken  up  into  its  se- 
parate particles,  each  being 
left  free  to  swing  as  a  simple 
pendulum  about  C.  The 
vertical  height  through  which 
0,  or  the  bullet,  will  rise  is 


The   particle  at 


tr 

% 

distance  r  will  rise  through 

T2     V2 

a  height  ji^-. 


Fig.  132. 


18 


274  MECHANICS  [CHAP. 

Let  m  be  the  mass  of  this  particle.     Then   the   rise  of  the 
centre  of  gravity  of  the  whole  system  of  free  particles  will  be 


Let  G  be  the  centre  of  gravity  of  the  rod,  distant  x  from  C, 
In  the  actual  oscillation  this  rises  through  a  vertical  height 

=  j  x  rise  of  0 

-is  .................................  » 

By  Huyghens'  principle  (1)  and  (2)  must  be  the  same.     And 
since  x  =  -^  —  (§  21),  we  have 

^     ^2 

L*'2       x    &       2mr 


v2 
and,  clearing  out  the  constant  factors  ^—  ,  Swi,  J!/, 


r       ^mr2 
so  that  L  - 


By  the  law  of  the  Simple  Pendulum,  the  time  of  oscillation  of 
the  rod  is  therefore 

fL     9 

27T  A  /  -  =  2-7T  A       - 

\  g  \  '  g  . 


where  M=  *£m,  the  total  mass  of  the  rod. 

The  problem  is  thus  reduced  to  the  calculation  of  the  quantity 
2mr2  for  the  rod.  Nowadays  this  is  easily  effected  for  a  body 
of  any  regular  shape  by  the  Integral  Calculus.  But  in  default 
of  modern  analysis  Huyghens  employed  the  following  ingenious 
device. 


XXVI] 


THE   COMPOUND   PENDULUM 


275 


253.  Let  CA  be  the  vertical 
imagine  a  triangular  flat  plate  ^ 
CAD  of  such  thickness  that 
the  mass  per  unit  length  of 
any  thin  slice  PQ  parallel  to 
the  base  is  equal  to  the  mass 
m  of  the  small  portion  of  the 
rod  opposite  to  it.  Then  if 
CP  =  PQ  =  r,  the  mass  of  PQ 
is  mr,  and  its  distance  below 
C  is  r. 

The  C.G.  of  the  triangular 
plate  is  at  a  distance  below  G 


rod.     Draw   AD  =  AC,   and 


\ 


Fi8-  133- 

But  this  is  the  value  of  L  for  a  uniform  rod.     The  C.G.  of  the 
triangle  is  known  to  be  at  a  vertical  distance  \GA  below  G. 

Thus  L  =  $.CA. 


EXAMPLES. 

1.  A  uniform  rod  suspended  by  one  end  makes  75  oscillations  per  minute. 
Find  the  distance  from  this  end  of  another  point  about  which  it  would  also 
make  75  oscillations  per  minute. 

2.  Find  the  length  of  a  uniform  rod  which  would  beat  seconds  when 
suspended  by  one  end. 


18—2 


CHAPTER  XXVII. 

D'ALEMBERT'S  PRINCIPLE. 

254.  THE  difficulties  which  Huyghens  surmounted  in  his 
Horologium  Oscillatorium,  1673,  for  the  special  problem  of  the 
Compound  Pendulum,  arise  with  ever  increasing  complication  in 
other  cases  of  the  motion  of  rigid  bodies. 

Seventy  years  afterwards  D'Alembert  (Traite  de  Dynamique, 
1743)  gave  a  general  principle  by  which  all  such  problems  may  be 
treated.  It  was,  perhaps,  not  so  much  a  new  principle,  as  an 
ingenious  device  for  reducing  the  problems  of  Rigid  Dynamics  to 
the  familiar  laws  already  established  for  the  equilibrium  of  forces. 

After  all,  a  rigid  body  consists  ultimately  of  separate  particles ; 
and  the  motion  of  each  of  these  is  determined  according  to  the 
Second  Law  of  Motion  by  the  resultant  of  the  forces  acting  on  it, 
including  the  reactions  from  its  neighbours  which  hold  it  in  position. 
The  difficulty  consists  in  the  number  of  particles  to  be  considered, 
and  the  unknown  nature  of  these  internal  reactions  between  them. 
D'Alembert  shewed  how  to  avoid  the  consideration  of  separate 
particles  and  internal  reactions. 

Consider  a  single  particle  of  mass  m.  It  will  be  subject  to 
(1)  its  weight  and  other  gravitational,  electric,  or  magnetic  attrac- 
tions and  repulsions,  and  perhaps  some  tensions  arid  pressures 
directly  applied  to  it  from  outside  the  body.  Call  these  the  im- 
pressed, or  external,  forces.  (2)  The  forces  which  bind  it  to  its 
neighbours,  and  hold  it  in  position  as  part  of  the  rigid  body.  Call 
these  the  internal  forces. 

Since  all  the  forces,  external  and  internal,  are  applied  to  a 
particle,  they  must  have  a  resultant.  Let  this  be  P. 


CHAP,  xxvn]  D'ALEMBERT'S  PRINCIPLE  277 

By  the  Second  Law  of  Motion  the  acceleration  a  of  the  particle 

p 

will   be  in  the  direction   of  this  resultant  and  equal   to  —  ;    so 

m 

that  ma  —  P. 

If,  then,  we  could  apply  to  the  particle  an  extra  force  ma, 
reversed,  i.e.  in  the  opposite  direction  to  P,  the  whole  set  of  forces 
acting  on  the  particle,  including  the  reversed  ma,  would  be  in 
equilibrium. 

Let  every  particle  in  the  body  be  treated  in  the  same  way, 
i.e.  let  a  force  be  applied  to  it  equal  to  its  mass  multiplied  by  its 
acceleration,  reversed  in  direction. 

Then  since  the  forces  acting  on  each  separate  particle  will  be 
in  equilibrium,  so  also  will  the  whole  system  of  forces  acting  on 
the  body.  This  system  consists  of  three  groups  : 

(1)  The  set  of  reversed  forces  of  the  type  ma.    Indicate  these 
by  2raa. 

(2)  The  set  of  internal  forces. 

(3)  The  impressed  or  external  forces  wherever  applied  to  the 
body.  ' 

Now  D'Alembert  points  out  that  the  second  group  consists  of 
pairs  of  equal  and  opposite  forces.  For  if  a  particle  A  exerts  an 
action  R  on  a  neighbouring  particle  B,  then  by  the  Third  Law  of 
Motion  B  exerts  an  equal  and  opposite  reaction  R  on  A.  When, 
therefore,  the  internal  forces  are  summed  for  the  whole  body,  they 
must  form  a  system  in  equilibrium  by  themselves,  and  exactly 
neutralize  each  other. 

It  follows  that  the  first  group  must  balance  the  third. 

Hence  the  system  of  the  reversed  forces  2ma  is  in  equilibrium 
with  the  forces  impressed  on  the  body  from  outside. 

255.  This  is  D'Alembert's  Principle.  It  is  applied  as  follows. 
Convenient  expressions  may  be  found  for  2ma  in  terms  of  (1)  the 
acceleration  of  the  centre  of  gravity  of  the  body,  and  (2)  the 
angular  accelerations  of  the  body  about  its  centre  of  gravity. 
Employing  these  expressions  we  write  down  the  conditions  of 
equilibrium  for  Sma  and  the  impressed  forces  according  to  the 
rules  of  Statics.  The  resulting  equations  are  sufficient  to  deter- 
mine the  motion  of  the  centre  of  gravity,  and  the  rotation  of  the 


278 


MECHANICS 


[CHAP. 


body  about  the  centre  of  gravity ;  and  thus  its  motion  is  com- 
pletely determined. 

The  problems  of  Rigid  Dynamics  are  often  extremely  difficult 
on  account  of  the  complexities  of  Solid  Geometry  and  Calculus 
required  for  their  solution.  Fortunately  some  of  the  most  im- 
portant are  also  the  simplest.  We  shall  give  here  one  or  two 
illustrations  of  D'Alembert's  Principle,  beginning  with  the  solution 
of  Huyghens'  problem  of  the  Compound  Pendulum,  for  the  sake 
of  some  experimental  results.  For  modern  developments  of  the 
subject  the  student  must  consult  treatises  on  Rigid  Dynamics. 


256.     Let  a  body  of  any  shape  swing  about  a  horizontal  axis 
The  compound         through  G.     Let 

r»l  iir\  f*  _  jr  ^^ 

IP 


Pendulum  by 
D'Alembert's 
Principle. 


0  be  the  inclina- 
tion to  the  ver- 
tical of  a  line  through  C  and  the 
centre  of  gravity  G. 

The  angular  velocity  of  CG 
about  C  is  the  rate  of  increase  of 
0  per  unit  of  time.  If  it  is  not 
constant,  we  proceed  exactly  as 
for  variable  velocities  in  a  straight 
line  (§71),  and  take  the  ratio  of 
the  very  small  angle  dO  (or  differ- 
ence of  0)  to  the  small  increment 
of  time  in  which  it  is  described. 
A  variable  angular  velocity  is  thus 
measured  by  the  limiting  value  of 

7/1 

the  fraction  -=-,  when   the  time  dt   is  made  indefinitely  small. 

Denote  this  by  0. 

As  in  the  case  of  rectilinear  velocities,  we  proceed  to  measure 
angular  accelerations  by  the  change  in  value  of  the  angular  velocity 
per  unit  of  time,  and  in  order  to  include  the  case  of  variable 
acceleration,  choose  an  indefinitely  small  time  in  which  the  velocity 
alters. 

Angular  acceleration  is  thus  measured  by  the  limiting  value 


Fig.  134. 


xxvn]  D'ALEMBERT'S  PRINCIPLE  279 

of  -.-  when  dt  is  taken  indefinitely  small.     Call  this  0.     In  the 
at 

72/3 

language  of  the  Differential  Calculus  it  is  written  -^  • 

Let  m  be  the  mass  of  the  particle  at  P  distant  r  from  the  axis. 
Then  if  angles  be  measured  in  circular  measure,  the  arc  described 
by  P  while  CP  moves  through  an  angle  <f)  is  r</>;  the  linear 
velocity  of  P  along  the  arc  is  r  x  angular  velocity  of  CP  =  r<j> ; 
and  the  acceleration  of  P  along  the  arc  is  r$. 

But  all  lines  from  C  to  particles  in  the  body  must  have  the 
same  angular  motion  about  G.  Thus  <J>  =  0  for  every  particle. 

We  can  now  express  D'Alembert's  Principle.  We  are  to  apply 
to  every  particle  m  a  force  equal  to  its  mass  x  its  acceleration 
reversed,  and  then  express  the  condition  that  all  these  forces 
balance  the  external  forces.  As  this  is  a  case  of  rotation  about 
an  axis  or  fulcrum,  the  latter  condition  must  be  that  the  sum  of 
the  moments  of  all  the  forces  about  the  axis  is  zero. 

Besides  the  acceleration  r$  along  the  tangent,  the  particle  has 
an  acceleration  towards  G  (==  r<£2,  §  77).  But  this  we  need  not 
consider,  since  it  will  have  no  moment  about  G.  The  moment  of 
the  force  mrQ  about  G  is  mrO  x  r.  The  moment  of  all  such  forces, 
reversed  in  direction,  for  every  particle  of  the  body  will  be 
—  2wr0  .  r  —  —  fewr2,  since  0  is  the  same  for  all  of  them. 

External  Forces.  The  external  forces  are  (1)  the  reaction  at 
the  axis  G,  and  (2)  the  weights  of  all  the  particles. 

The  former  has  no  moment  about  the  axis. 

The  latter  are  equivalent  to  the  whole  weight  of  the  body  supposed 
collected  at  G.  Let  the  distance  GG,  from  the  axis  to  the  centre 
of  gravity,  be  h,  and  let  M  be  the  mass  of  the  body.  Then  its 
weight  is  Mg,  and  its  moment  about  the  axis  is  Mg .  h  sin  6.  This 
is  to  be  counted  negative  since  it  is  in  the  clockwise  direction. 

By  D'Alembert's  Principle 

—  02rar2  —  Mgh  sin  6  —  0. 

*         Mgh     .     - 
.*.  0  =  -     '    .  sin  0. 


As  in  the  case  of  the  Simple  Pendulum,  let  us  suppose  that 


280  MECHANICS  [CHAP,  xxvn 

6  is  a  small  angle  such  that  its  circular  measure  may  be  put  for 
its  sine.     Then 


ti 

V   =   --  ^  -  o  '        } 

Zmr2 

or,  the  angular  acceleration  is  proportional  to  the  angular  dis- 
placement from  the  position  of  rest,  and  tends  towards  it. 

But  this  is  the  law  of  Simple  Harmonic  Motion  (§  208). 

The  pendulum  will  therefore  describe  a  Simple  Harmonic 
Vibration  of  periodic  time 


ZTT  A/  -jTr-r- 
V   Mgh 

This  is  Huyghens'  result. 


EXAMPLES. 

1.  A  compound  pendulum  with  mass  J/,  moment  of  inertia  (§  257)  about 
the  axis  /,  distance  of  centre  of  gravity  from  axis  h,  is  released  when  the  line 
through  the  axis  and  the  centre  of  gravity  is  inclined  a  to  the  vertical.     Shew 
that  the  angular  velocity  o>  when  this  line  is  inclined  6  to  the  vertical  is 
given  by 

2     2Mgh ,       A 
or =  — j—  (cos  0  -  cos  a). 

(Equate  the  energy  (§  260)  of  the  pendulum  to  the  work  done.) 

2.  Shew  that  the  angular  velocity  of  the  simple  equivalent  pendulum  of 
length  Z,  released  simultaneously  at  the  same  inclination  a,  is  given  by 

£»2=  ~  (COS  6  -  COS  a). 

3.  Deduce  the  formula  for  the  length  of  the  simple  equivalent  pendulum 
by  comparing  the  results  of  questions  1  and  2. 


CHAPTER  XXVIII. 


MOMENT  OF  INERTIA. 

257.  THE  quantity  Smr2,  which  occurs  in  all  problems  con- 
nected with  rotation  about  an  axis,  was  called  by  Euler  the 
Moment  of  Inertia  of  the  body  about  the  axis.  It  is  the  sum  of 
the  products  of  each  element  of  mass  by  the  square  of  its  distance 
from  the  axis.  The  following  Table  gives  its  value  for  the 
principal  regular  figures  as  calculated  by  the  Integral  Calculus. 
We  shall  see  later  how  Moments  of  Inertia  may  be  determined 
experimentally. 


Table  of  Moments  of  Inertia. 
The  moment  of  inertia  of 

(1)  A  rectangle  whose  sides  are  2a,  26 
about   an   axis   through    its    centre    in    its 

plane    perpendicular    to    the    side    2a 

about  an  axis  through  its  centre  perpendicular 

to  its  plane 

(2)  A  rectangular  block,  sides  2a,  26,  2c, 
about  an  axis  through  its  centre  perpendi- 

cular to  the  sides  2a,  26 

(3)  An  ellipse,  semi-axes  a  and  6 
about  the  major  axis  a 


x  — 


a2 
3' 


=  mass  x 


=  mass  x 


62 


mass  x  — 


about  the  minor  axis  6 

about  an  axis  through  its  centre  perpendi- 
cular to  its  plane 


4' 
2 


=  mass  x  - 


mass  x 


a 
4' 

a*  +  6* 

4 


282  MECHANICS  [CHAP. 

In   the  particular  case  of  a  circle,  radius  a,  the  moment  of 

inertia  about  a  diameter  =  mass  x  — ,  and  about  a  perpendiculai 

a? 
to  its  plane  through  the  centre  =  mass  x  —  . 

(4)  An  ellipsoid,  semi-axes  a,  b,  c 

62  +  c2 

about  the  axis  a  =  mass  x . 

5 

In  the  particular  case  of  a  sphere  of  radius  a,  the  moment  of 

inertia  about  a  diameter  =  mass  x  -V-  . 

o 

Dr  Routh,  from  whose  treatise  on  Rigid  Dynamics  this  table 
is  taken,  gives  an  easy  rule  for  remembering  it. 

The  moment  of  inertia  about  an  axis  of  symmetry 

sum  of  squares  of  perpendicular  semi-axes 

=  mass  x  ±          — ii-     — — - 

3,  4,  or  5 

The  denominator  is  to  be  3,  4,  or  5  according  as  the  body  is 
rectangular,  elliptical  (including  circular),  or  ellipsoidal  (including 
spherical). 

(5)  A  cylinder,  radius  a,  length  21, 

about  its  axis  =  mass  x  -=• , 

about  an  axis   through   its   centre  perpen-  2      , 

dicular  to  its  axis  =  mass  x  t  —  +  ^  J . 

258.  The  usefulness  of  this  table  is  greatly  extended  by  the 
following  theorem. 

The  moment  of  inertia  about  any  axis  is  equal  to  the  moment 
of  inertia  about  a  parallel  axis  through  the  centre  of  gravity  plus 
the  moment  of  inertia  of  the  whole  mass,  collected  at  its  centre  of 
gravity,  about  the  original  axis. 

Let  the  axis  be  perpendicular  to  the  plane  of  the  paper  at  0. 
Let  a  parallel  axis  through  the  centre  of  gravity  cut  this  plane 
in  G. 

Let  OM  =  x;  GM=y. 

Let  ra  be  the  mass  situated  on  a  line  through  any  point  P 
perpendicular  to  the  plane  of  the  paper. 
MN=x; 


XXV  III]  LINEAR   AND   ANGULAR   MOTION 

Then  moment  of  inertia  about  axis  0 

=  2m .  OP2  =  2m  [(x  +  x?  +  (y  + 


283 


Fig.  135. 

But  since  G  is  the  centre  of  gravity 

2ma?  =  0    and    2my=0,     (§21). 
And  06r  is  the  same  for  each  term  of  2m  .  OG.     Therefore 


if  M  is  the  total  mass.     2m$Pa  is  the  M.I.  (Moment  of  Inertia) 
about  the  axis  G. 
Hence 

M.I.  about  the  axis  0  =  M.I.  about  axis  G  +  M  x  OG*. 

259.  It  is  easy  to  see  that  Moments  of  Inertia  on  the  one 
hand  and  the  Statical  Moment  of  the  external  forces  about  the 
axis  on  the  other  play  the  same  parts  with  regard  to  rotations  as 
masses  and  impressed  forces  with  respect  to  motions  of  translation. 
Thus  for  the  latter 

Mass  x  acceleration  =  Impressed  Force, 
or  M  a  =  P. 

For  the  compound  pendulum 

2mr2  x  S  =  -  Mg  .  h  sin  9, 


284 


MECHANICS 


[CHAP. 


i.e.,     Moment    of    Inertia  x  angular    acceleration  =  Moment    of 
Impressed  Forces. 

In   general   if  /  is   the   moment   of  inertia,   a   the   angular 
acceleration,  and  G  the  moment  of  the  impressed  forces, 


The  law  expressed  by  this  formula  is  the  exact  analogue,  for 
rotations,  of  the  Second  Law  of  Motion,  expressed  by  its  formula 
Ma  —  P,  for  motions  of  translation.  The  one  is  as  fundamental 
for  rotation  as  the  other  is  for  translation. 

In  the  case  of  constant  angular  acceleration,  kinematical 
formulae  may  be  found  for  the  angular  velocity  acquired,  o>, 
and  the  angle  turned  through,  0,  precisely  similar  to  those  for 
the  velocity  and  distance  travelled  in  §111. 

260.  Comparison  of  formulae  for  linear  and  angular  motion 
under  constant  acceleration. 


Kinematical  Formulae. 
Linear.  Angular. 


at2 

5  =  T 

v- 
-2=as 


Dynamical  Formulae. 


Mv  =  Pt 
(Momentum  =  Impulse) 


(Energy  =  Work  done) 


G 


(Angular  Momentum 

=  Moment  of  Impulse) 


(Energy  =  Work  done) 


xxvm] 


LINEAE   AND   ANGULAR   MOTION 


285 


The  expression  Iw,  which  corresponds  to  momentum  in  linear 
motion,  is  called  the  angular  momentum,  or  moment  of  the 
momentum  of  the  rotating  body.  For  the  particle  m  distant  r 
from  the  axis  has  a  velocity  v  =  cor  along  the  tangent.  Its 
momentum  is  mv  =  mcor.  The  moment  of  this  momentum  about 
the  axis  is  ma>r  x  r,  and  the  moment  of  the  momentum  for  the 
whole  body  is  %ma)r  x  r  =  co^mr2  =  I .  co. 

Again,  -^-  is  the  Energy.     For  the  energy  of  the  particle  m  is 

-^— =  JwwV2.     The  energy  of  the  whole  body  is  therefore 

"  "      co2 


^T-T*""*-7^ 

Lastly,  GO  is  the  work  done  by  the  couple  of  moment  G  in 
turning  through  an  angle  whose  circular  measure  is  0. 


Fig.  136. 

For  let  AB  be  the  arm,  and  let  it  turn  about  any  point  C  through 
the  small  angle  6.     The  work  done  by  the  forces  is 

PxAA'  +  PxBB' 


=  PxAB.0 

=  moment  of  couple  x  6  =  G  .  0. 

261.  To  make  the  analogy  between  the  equations  of  linear 
and  angular  motion  clearer,  let  us  consider  a  simple  case. 

A  heavy  wheel  or  disc  on  an  axle  0  of  radius  a  is  set 
rotating  by  a  cord  coiled  round  the  axle  and  pulled  with  a  steady 
force  P.  To  find  the  motion. 


286 


MECHANICS 


[CHAP. 


Every  particle  of  the  wheel  must    have    the   same   angular 
acceleration  about  0.    Let  this 
be  a.     Then  by  D'Alembert's 
Principle  and  taking  moments 
about  0 

—  2  mar  x  r  +  P  .  a  =  0. 
Pa   _Pa 

Since  this  is  constant,  the 
angular  velocity  at  time  t  is 

a>  =  at  =  ^.t,  Fig.  137. 

and  the  angle  turned  through  by  the  wheel  from  rest  is 


At  the  beginning  of  Statics  we  found  that,  as  Leonardo 
perceived,  the  Moment  of  a  force  was  the  proper  measure  of  its 
Statical  tendency  to  turn  a  body  round  a  fulcrum.  It  now  appears 
that,  when  rotation  ensues,  the  Moment  is  still  the  proper  measure 
of  the  efficacy  of  the  force  in  producing  angular  momentum.  So 
that  in  all  circumstances  what  we  have  called  the  Torque,  i.e.,  the 
twisting  or  turning  effect  of  a  system  of  forces,  is  to  be  measured 
by  their  Moment. 

262.  Next,  let  the  cord,  instead  of  being  pulled  with  constant 
force,  hang  vertically  and  support  a  mass  m.  There  are  now  two 
equations  of  motion,  one  for  the  rotating  wheel,  and  one  for  the 
linear  motion  of  m.  Let  T  be  the  tension  of  the  cord ;  a  the 
angular  acceleration  of  the  wheel ;  a'  the  linear  acceleration  of  m. 

As  before,  for  the  wheel ; — 

a=  — 

For  m ; — 


'Ill 


To  determine  the  unknown  tension  T  there  is  the  geometrical 
relation 


SXVIII]  LINEAR   AND   ANGULAR   MOTION  287 

since  the  acceleration  of  m  is  the  same  as  that  of  the  rim  of  the 
axle. 

Ta*  T 

Hence  —j—  —  g  --  , 

1  m  ' 


rr  fa*      *  \  j  rr       ™gl 

.*.    rl  (-=  +  -    =  or,  and  T=  T~  -  —  -  , 

\I      m)  J-fflitt* 

aT  mga 

so  that                              a.  =  -=r-  =  ,  —  ?  , 

/       «*a»  +  / 

and  the  motion  is  known. 


EXAMPLES. 

1.  Calculate  in  foot-tons  the   energy   of  a   10-ton   flywheel   8   feet  in 
diameter,  revolving  100  times  a  minute,  assuming  that  the  whole  mass  is 
collected  in  the  rim. 

2.  In  a  certain  engine  the  piston  diameter  is  8  inches,  the  mean  steam 
pressure  80  Ibs.  per  square  inch,  and  the  length  of  stroke  3  feet.     What  must 
be  the  mass  of  a  flywheel,  supposed  collected  in  the  rim,  which  is  to  have 
6  feet  diameter,  and  store  at  least  100  times  the  energy  supplied  in  each 
stroke  when  running  at  200  revolutions  per  minute  ? 

3.  The  inner  and  outer  radii  of  the  rim  of  a  flywheel  running  at  60 
revolutions  per  minute  are  7  and  8  feet  respectively,  and  its  mass  is  10  tons. 
Neglecting  the  spokes,  find  its  energy  in  foot-tons.     How  long  would  it  take 
a  50  H.  P.  engine  to  get  it  up  to  this  speed  ? 

4.  In  an  Atwood's  machine  the  weights  are  each  8  ounces,  the  rider 
1  ounce,  and  the  pulley  weighs  2  ounces.     Find  the  acceleration  (1)  if  the 
pulley  is  a  ring  with  spokes  of  negligible  mass  ;  (2)  if  the  pulley  is  a  uniform 
flat  disc. 

5.  A  pendulum  consists  of  a  flat  rod,  40  inches  long  and  weighing  2  Ibs., 
with  a  sphere  of  6  inches  diameter,  weighing  10  Ibs.,  rigidly  fixed  upon  it  so 
that  the  lower  end  of  the  rod  projects  2  inches  beyond  the  sphere.     How 
many  beats  will  it  make  per  minute  ? 

6.  Find  the  time  of  swing  of  a  cube  about  one  of  its  edges  which  is 
horizontal  and  of  length  2a.    What  must  the  length  of  the  edge  be  so  that  the 
cube  may  beat  half  seconds  ? 


288  MECHANICS  [CHAP,  xxvm 

7.  The  handle  of  a  wheel  and  axle  is  let  go  just  as  a  bucket  full  of  water 
weighing  60  Ibs.  reaches  the  top  of  a  well  18  feet  deep,  and  the  bucket  gets  to 
the  bottom  again  in  6  seconds.     If  the  axle  is  6  inches  in  diameter,  find  the 
moment  of  inertia  of  the  wheel  and  axle,  neglecting  friction. 

8.  A  solid  sphere  rolls  down  a  rough  inclined  plane  (angle  a)  without 
slipping.     Shew  that  its  acceleration  is  f  g  sin  a. 

9.  A  hoop  rolls  down  a  roof  sloping  30°  to  the  horizon  for  a  distance  of 
16  feet  from  rest.     Shew  that  its  velocity  is  the  same  as  if  it  had  fallen 
vertically  through  4  feet. 

How  does  this  agree  with  Galileo's  principle  of  motion  on  an  inclined 
plane  ? 

10.  It  has  been  proposed  to  draw  energy  for  industrial  purposes  from 
the  earth's  energy  of  rotation  on  her  axis,  by  utilising  the  ebb  and  flow  of 
the  tides.     Taking  the  mass  of  the  earth  as  T35  x  1025  Ibs.,  and  her  radius  as 
4000  miles,  shew  that,  including  what  is  wasted  by  friction  of  the  tides,  she 
could  supply  one  million  horse-power  continuously  for  about   ll£  million 
million  years. 


CHAPTER  XXIX. 

EXPERIMENTAL  DETERMINATION  OF  MOMENTS  OF  INERTIA. 

263.     THE  wheel  is  mounted  on  a  horizontal  or  vertical  axle 
and  set  in  motion  by  a  weight  hanging  from  a 

Moment  of  it  . 

inertia  of  a  cord  coiled  round  the  axle.  A  loose  loop  at  the 
other  end  of  the  cord  is  passed  over  a  pin  on  the 
axle.  If  the  axle  is  vertical,  the  cord  must  be  carried  over  a  light 
fixed  pulley. 

The  work  done  by  the  weight  in  descending  a  measured 
distance  to  the  ground  is  equal  to  the  energy  of  the  wheel  and 
of  the  descending  weight,  together  with  what  has  been  expended 
in  overcoming  friction. 

Let  F  be  the  work  used  up  in  overcoming  friction  during  one 
turn  of  the  wheel ;  m  the  mass  of  the  weight ;  /  the  moment  of 
inertia  of  the  wheel ;  n  the  number  of  revolutions  up  to  the 
moment  when  the  weight  reaches  the  ground  and  the  cord  slips 
off  the  pin;  «  the  angular  velocity  at  that  moment.  Let  the 
wheel  make  n'  more  revolutions  before  coming  to  rest. 

(1)  To  find  o>.  A  strip  of  smoked  paper  is  fastened  round 
the  rim  of  the  wheel  by  gumming  the  ends.  A  tuning  fork 
making  a  known  number  of  vibrations  per  second — say  100 — is 
held  in  a  clamp,  and  carries  a  bristle  or  metal  pointer  on  one 
prong.  Just  as  the  weight  reaches  the  ground,  the  fork  is  struck 
with  a  rubber  cork  mounted  on  a  brass  wire,  and  the  pointer 
pressed  lightly  for  a  moment  against  the  smoked  paper,  by  a 
slight  turn  of  the  clamp  on  ifcs  stand.  If  the  fork  is  set  so  as  to 
vibrate  parallel  to  the  axle,  the  result  will  be  a  trace  on  the 
c.  19 


290 


MECHANICS 


[CHAP. 


Fig.  138. 

smoked  paper  consisting  of  a  number  of  waves.     Measure   the 
whole   length   occupied   by  as   many   of  these   as   can   be   seen 


XXIX]  THE   FLYWHEEL  291 

distinctly,  and  count  the  waves.  Let  the  length  from  any  point 
on  the  first  wave  to  the  corresponding  point  on  the  pth  wave  be  /. 
Then  the  length  of  one  wave  is  l/p,  and  since  100  waves  pass  the 
fork  in  one  second,  the  velocity  of  the  rim  must  have  been 
100  x  l/p.  Measure  the  diameter  of  the  wheel  by  the  calipers, 
and  hence  find  the  radius  R. 


Then  wJB  =  100x  J/p  ........................  (1). 

Let  z  be  the  distance  fallen  through  by  the  weight.     This 
may  be  measured  before  the  wheel  is  allowed  to  start.     Then 

mgz  —  work  done  by  weight 

=  energy  of  wheel  +  energy  of  weight  -f  loss  in  friction 


(2), 


where  v  =  the  velocity  of  the  weight  =  aa>. 

When  the  wheel  comes  to  rest  after  nr  more  turns,  the  whole 
of  the  work  has  been  absorbed  by  friction,  except  the  energy  of 
the  weight  when  it  was  stopped  by  the  ground. 


Thus  mgz  =  (n  +  ri)F+  ..................  (3). 


From  (3)  F  may  be   found,  and  since  a  is  small,  the  term 
— - —  may  generally  be  neglected,  so  that 


n  +  n 


o)  is  known  from  (1),  and  /  can  be  found  from  (2). 

As  a  check  on  the  counting,  see  whether  z  =  2?ra .  n.  It  is 
better  to  measure  z  directly,  instead  of  calculating  it  from  this 
formula,  to  avoid  errors  arising  from  uneven  winding  and  stretching 
of  the  string. 

264.     A  heavy  cylinder  is  clamped  to  one  end  of  a  brass 
The  Torsion        wire.      The   other   end   of  the   wire   is   held   in 

Pendulum.  ft    fixed    clamp< 

19—2 


292 


MECHANICS 


[CHAP. 


The  cylinder  carries  a  cross  arm,  to  the  ends  of  which  small 
weights  may  be  attached, 
and  a  pointer  moving  over 
a  flat  circular  scale,  so  that 
the  angle  turned  through 
by  the  cylinder  can  be  read. 

When  the  cylinder  is 
turned  about  the  vertical 
wire,  the  twisted  wire  exerts 
a  couple  tending  to  bring 
it  back  to  the  original  posi- 
tion. The  moment  of  this 
couple  is  proportional  to 
the  angle  turned  through. 
(§  210.) 

Turnthecylinderthrough 
a  small  angle  and  set  it  free. 

Let  6  be  the  angular 
deflection  at  any  moment; 
0  the  angular  acceleration ; 
/  the  M.I.  of  the  suspended 
system  about  the  vertical. 

Then  by  D'Alembert's 
Principle,  as  in  the  case  of 
the  Compound  Pendulum, 


Tig.  139. 


M.I.  x  angular  acceleration  =  —  (moment  of  twisting  couple). 

Let  G  be  the  torque,  or  moment  of  the  couple  required  to 
twist  the  wire  through  a  unit  angle.  Then  G6  is  the  couple  for 
a  deflection  0. 


and  the  motion  will  be  Simple  Harmonic  with  a  period  2-Tr*  /  -~. 

In  fact  the  best  proof  that  the  couple  is  proportional  to  the 
angle  of  torsion  is  that  the  vibrations  are  simple  harmonic  vibra- 
tions. Verify  that  this  is  so,  by  timing  20  swings  for  amplitudes 


XXIX]  THE   TORSION    PENDULUM  293 

of  5°,  10°,  20°,  30°  on  each  side  of  zero.  The  time  of  swing  will 
be  found  to  be  the  same. 

Observe  accurately  the  time  of  50  complete  oscillations.  It  is 
best  to  note  the  instants  of  passing  the  zero,  not  the  moment  of 
coming  to  rest  on  one  side  or  the  other,  since  the  exact  moment 
can  be  more  sharply  fixed  when  the  pointer  is  moving  rapidly. 

From  the  duration  of  50  swings  find  the  time  of  one  swing. 
Let  this  be  2\.  Then 


Hang  from  the  ends  of  the  cross  arm  two  equal  small  weights, 
having  first  found  their  masses,  m,  by  weighing  them  on  a  sensitive 
balance. 

Let  TZ  be  the  time  of  swing  with  the  weights  added.  Measure 
I  the  distance  of  either  small  weight  from  the  vertical  axis.  The 
moment  of  inertia  of  the  system  is  now  /+2m/2;  the  couple  G, 
depending  only  on  the  wire,  is  unaltered.  So  that 


Equations  (1)  and  (2)  determine  /  and  G. 
Thus  - 


2nd 


The  difference  T?—Tf  must  not  be  very  small,  or  a  slight 
error  in  determining  Tl  and  Tz  will  make  a  great  difference  in  the 
value  of  /.  If,  however,  larger  weights  are  employed,  they  cannot 
be  treated  as  particles,  and  their  moment  of  inertia  about  the  axis 
must  be  expressed  by  adding  to  2ml2  the  sum  of  their  moments 
of  inertia  about  vertical  axes  through  their  own  centres  of  gravity. 

265.     From  (1) 


29  4  MECHANICS  [CHAP. 

Having  once  found  G  for  a  particular  wire,  we  may  use  the 
wire  to  determine  by  a  single  observation  the  moments  of  inertia 
of  any  objects  that  can  be  clamped  to  it.  For  if  K  be  the  moment 
of  inertia  of  the  object,  the  time  of  vibration  when  it  is  suspended 

by  the  wire  will  be  T=%TT  *  /TT-     Observing  T  we  can  at  once 


The  student  should  in  this  manner  verify  some  of  the  formulae 
in  §  257. 

266.     The  important  constant  called  the  modulus  of  torsion 

is  the  moment  of  the  couple  required  to  twist  a 

?ora?onof  i  wire      un^  length  of  the  wire  through  a  unit  angle,  i.e. 

by  Maxwell's  one  radian.     Let  this  be  denoted  by  r.    Then  the 

Needle.  .  .  J 

couple  required  to  twist  one  end  of  a  wire  of  length 


I  through  an  angle  6  will  be  j  6,  since  each  unit  of  length  is  only 

n 
twisted  through  j  . 

We  might  measure  the  length    of  the  wire   in  experiment 
§  264,  and  determine  T  from  the  equation 


But  it  may  be  found  much  more  accurately  by  means  of 
a  piece  of  apparatus  devised  by  Clerk  Maxwell  for  use  in  the  study 
of  the  viscosity  of  gases.  Figure  140  shews  the  instrument,  which 
is  known  as  Maxwell's  Needle. 

The  "needle"  is  a  hollow  brass  cylinder  provided  with  a  central 
clamp  for  suspension  by  the  wire.  The  other  end  of  the  wire  is 
held  in  a  clamp  supported  at  the  head  of  a  hollow  vertical  brass 
pillar,  with  a  torsion  head,  i.e.  the  top  of  the  pillar,  bearing  the 
clamp,  can  be  turned  round  so  as  to  adjust  the  position  of 
equilibrium  of  the  needle.  There  is  a  glass  case,  supported  on 
levelling  screws,  to  protect  the  needle  from  disturbance  by  currents 
of  air. 

Inside  the  needle  slide  four  brass  cylinders,  each  one-quarter 
of  its  length.  Two  of  these  are  hollow,  and  of  equal  weight  ;  the 
other  two  are  filled  with  lead,  and  are  also  of  equal  weight.  There 


xxix] 


MAXWELL'S  NEEDLE 


295 


Fig.  140. 

is  a  scale  on  the  needle  by  which  the  inner  cylinders  can  be  set 
accurately  in  position. 

To  find  the  modulus  of  torsion  of  a  wire,  the  needle  is 
suspended  by  it,  and  the  cylinders  are  slid  into  position  with  the 
two  heavy  ones  on  the  inside,  and  the  hollow  ones  at  the  ends. 
The  needle  is  set  vibrating,  and  the  time  of  oscillation  TV  is 
determined. 

The  cylinders  in  the  needle  are  then  rearranged  so  that  the 
heavy  ones  are  at  the  ends ;  and  the  time  of  oscillation  Tz  is  again 
determined. 

To  assist  in  finding  Tt  and  Ta  with  the  greatest  accuracy  the 
needle  is  provided  with  a  small  plane  mirror  at  its  centre.  Opposite 
this  a  reading  telescope  with  scale  is  set  up  so  that  the  image  of 
the  scale  formed  by  the  mirror  is  seen  sharply  in  the  telescope. 
The  torsion  head  can  be  turned  till  the  zero  at  the  middle  of  the 
image  of  the  scale  coincides  with  a  vertical  cross  wire  in  the  focus 
of  the  telescope.  When  the  needle  is  oscillating,  the  scale  will 


296  MECHANICS  [CHAP. 

appear  to  cross  the  field  of  view  of  the  telescope,  and  the  instant 
of  the  passage  of  the  zero  can  be  fixed  with  great  exactness.  By 
observing  the  time  of  10  swings,  taking  the  time  from  a  laboratory 
clock  ticking  seconds,  or  a  chronometer  ticking  half  seconds,  the 
time  of  swing  may  be  first  found  to  within  '05  of  a  second.  From 
this  the  approximate  time  at  which  the  hundredth  swing  will 
occur  is  calculated.  The  needle  is  left  swinging,  and  when  the 
proper  moment  approaches,  the  observer  takes  his  station  and 
records  the  actual  instant  of  the  hundredth  passage.  From  this 
a  much  closer  value  of  the  time  of  swing  may  be  found  ;  and  if 
necessary,  this  may  be  employed  in  the  same  way  to  allow  the 
observation  of  a  still  greater  number  of  swings. 

267.     Let   m,  m'  be   the  masses  of  the  loaded  and  hollow 

Theory  of  the      cylinders  respectively;  c  the  length  of  a  cylinder; 

instrument.        j  the  M  L  of  the  empty  needle  ;  /j  and  /2  those  of 

the  loaded  and  hollow  cylinders  about  vertical  axes  through  their 

centres  of  gravity. 

Let  K  be  the  M.I.  of  the  system  with  the  loaded  cylinders  in 
the  middle  ;  K  +  k  the  M.I.  when  the  heavy  cylinders  are  at  the 
ends. 

Then  r  =  2 


Thus 


'.  k.-  = 

T 


and 


rpz  __  ^T2- 

It  remains  to  find  k.     By  §  258 

K  =  I  +  27,  +  2/2  +  2m  (^Y  +  2m'  ( ^ ) 


and         K  +  /j  =/+  27,  +  2/2+  2m  (%C)  +  2m'  [|j  . 


xxix]  MAXWELL'S  NEEDLE  297 

7r2c2  (m  -  m')  I 
whence  T  =  — ^ — ,r        . 

•»§  •"•*! 

The  length  I  of  the  wire  from  clamp  to  clamp  is  measured 
with  a  beam  compass  or  steel  tape. 

Having  found  this  constant  for  a  particular  wire,  Maxwell 
employed  the  wire  for  suspending  flat  discs  in  gases  at  various 
pressures,  and  from  the  observed  damping  of  their  oscillations  he 
deduced  the  effects  of  the  viscosity  of  the  gases  on  the  surfaces  of 
the  discs. 

268.  The  method  of  altering  the  M.I.  by  adding  or  shifting 
bodies  of  known  mass  and  form,  whose  moments  of  inertia  can  be 
calculated,  and  then  observing  the  time  of  swing,  is  employed  in 
many  physical  measurements.     Thus  in  determining  the  strength 
of  the  Earth's  magnetic  field  by  the  Kew  pattern  of  magnetometer, 
the  M.I.  of  the  suspended  magnet  is  found  by  adding  to  it  a  small 
brass  cylinder.     As  brass  is  non-magnetic,  the  couple  exerted  on 
the  system  by  the  earth's  field  is  not  altered. 

The  M.I.  of  themag.net  in  a  Ballistic  Galvanometer  is  some- 
times found  in  the  same  manner,  but  more  usually  by  the  method 
employed  with  the  Ballistic  Pendulum  of  the  following  article. 

269.  The  Ballistic  Pendulum  figured  on  p.  298  is  merely  a  mas- 

,      sively  constructed  Balance.   The  heavy  framework 

Determination  of  •/  J 

the  velocity  of  a        representing  the  beam  rests  on  steel  knife  edges 

bullet  by  the  &  . 

Ballistic  m  steel  V-shaped  cups.     The  heavy  weight  hang- 

Pendulum.  .  ,     .  .     r.  .         ,     ,      f     '       ,.   °   . 

mg  below  is  the  gravity  bob  for  adjusting  the 
distance  of  the  centre  of  gravity  from  the  knife  edges.  Above  are 
seen  a  large  weight  and  a  small  metal  flag  for  coarse  and  fine  ad- 
justment of  the  pointer  to  zero.  In  this  instrument  the  readings 
may  also  be  taken  by  means  of  the  image  of  the  curved  scale  seen 
against  a  horizontal  fixed  pointer. 

The  two  heavy  cylinders  suspended  from  the  beam  may  be 
filled  with  shot  and  adjusted  to  equal  weights.  They  hang  by 
knife  edges  resting  in  grooves  which  serve  to  graduate  the  beam. 
They  can  thus  be  set  at  equal  measured  distances  from  the  axis, 
and  as  they  do  not  rotate  with  the  beam,  but  always  hang 


298 


MECHANICS 


[CHAP. 


i — -=9 


Fig.  141. 


vertically,  the  change  they  make  in  its  M.I.  is  merely  the  product 
of  their  masses  into  the  squares  of  the  distances  of  their  points  of 
suspension  from  the  central  knife  edge. 

Moreover,  a  horizontal  shift  of  these  weights  does  not  change 
the  distance  of  the  C.G.  from  the  axis,  so  that  the  couple  tending 
to  restore  equilibrium  is  unaltered.  They  thus  afford  a  means  of 
both  measuring  the  M.I.  and  adjusting  it  to  a  convenient  value, 
without  otherwise  changing  the  circumstances. 

Above  the  centre  of  the  beam  is  an  upright  carrying  a  heavy 
metal  disc  on  which  a  blow  may  be  struck ;  or  a  block  of  wood 
may  be  clamped  there  to  receive  a  bullet  from  a  revolver.  Such 
a  blow  will  cause  the  instrument,  previously  at  rest,  to  swing 
through  a  certain  angle,  and  then  continue  oscillating.  The 
object  is  to  find  from  its  movements  the  momentum  communicated 
to  it  by  the  blow.  From  this  the  velocity  of  the  striking  object 
may  be  found  if  its  mass  is  known. 


XXIX] 


THE   BALLISTIC    PENDULUM 


299 


270.     Let  /  be  the  M.I.  of  the  pendulum  about  the  knife 
edges. 

Theory  of  the  ,  .  . 

instrument.  <o  =  the  angular  velocity  with  which  it  begins 

to  swing. 

m  =  the  mass  of  the  bullet,  and  v  its  velocity. 
k  =  distance  from  knife  edge  to  line  of  fire  of  bullet. 

Then 

Moment  of  momentum  of  pendulum 

=  moment  of  impulse  of  bullet  about  the  axis. 
la)  =  mvk, 

Ia> 
V=—j-    (1). 

mk 


and 


(Strictly,  /,  the  moment  of  momentum  of  the  pendulum,  should 
include  that  of  the  bullet  embedded  in  it.  This  may  be  partially 
allowed  for  by  taking  the  time  of  swing  after  the  bullet  has  been 
fired.  But  in  any  case  the  mass  of  the  bullet  is  too  small  to  make 
any  appreciable  difference.) 

(1)     To  find  /. 

Let  M  be  the  mass  of  the  pendulum ;  OG  =  h  the  distance  of 
the  C.G.  from  the  axis. 

Observe  the  time  of  oscillation  of  the  pendulum.  Let  it 
be  T. 


Fig.  142. 


Then 


T  = 


Mgh 


.(2). 


300 


MECHANICS 


[CHAP. 


Next,  place  a  small  weight  of  mass  ra'  in  one  of  the  cylinders  ; 
let  I  be  the  distance  of  a  cylinder  from  the  axis,  and  let  a  be  the 
steady  deflection  from  the  horizontal  when  the  pendulum  comes  to 
rest. 

Then  by  moments  about  0 

Mg  .  h  sin  a  =  ing  .  I  cos  a    .................  (3). 

From  (2)  and  (3), 


(2)     To  find  ft>. 

The  pendulum  will  swing  aside  when  the  blow  is  struck  till  its 
energy  is  exhausted  by  the  work 
done  against  gravity.  Let  0  be 
the  angle  through  which  it  swings 
before  it  comes  to  rest  for  the  first 
time. 

The  C.G.  rises  through  a  vertical 
height 

GH  =  OG  -  OH  =  h  (1  -  cos  £). 

The  work  done  against  gravity 
=  Mg.h(l-cosj3), 


Fig.  143. 


.  sin|  =  ^sinf,by  (2). 


4-7T  £ 

a  X  ~T  '  Sm  "2 


=  2Mgh  sin2  £ , 

and  a)  =  2 

Hence  finally 


mk     47r2 

m'gl.T 


mk 


.  cot  a .  sin     . 


271.     The  experiment  should  be  conducted  in  the  following 
order. 


XXIX]  THE   BALLISTIC    PENDULUM  301 

Adjust  the  balance  to  zero  after  the  wooden  block  is  clamped 
on. 

Observe : — 

(1)  The  steady  deflection  a  when  the  known  weight  m'  is 
placed  in  one  of  the  cylinders,  using  both  the   pointer  and  the 
reflected  scale. 

(2)  The    distance   of  cylinder-suspension  from   the   knife 
edge  =  I. 

(3)  The  value  of  the  first  swing  when  the  bullet  strikes 
=  /3.     Be  sure  that  the  Balance  is  at  rest  before  tiring.     Practise 
observing  the  first  swing  when  a  blow  is  struck,  before  trying  with 
the  bullet. 

(4)  The  distance  from  axis  of  line  of  fire  of  bullet  =  k. 

(5)  The  time  of  vibration  after  the  bullet  has  been  fired 
in  =  21 

(6)  The  mass  of  the  bullet  =  m. 

From  these  observed  values  v  is  determined  by  the  formula. 

272.  The  moment  of  inertia  I  might  have  been  found  by 
shifting  the  two  cylinders  to  another  distance  I'  from  the  axis,  and 
again  observing  the  time  of  vibration  Tf.  Weigh  one  of  the 
cylinders,  and  let  its  mass  be  M'.  Then,  if  K  be  the  M.I.  of  the 
frame  without  the  cylinders, 


Mgh 


Mgh 

whence  K  is  known,  and  I=K+  2M'l2,  may  be  calculated. 

We  have  preferred  to  use  the  steady  deflection  by  a  known  weight, 
because  this  method  is  a  strict  mechanical  analogue  to  the  practice 
with  the  Ballistic  Galvanometer  in  electrical  measurements. 


302  MECHANICS  [CHAP.  XXIX 


EXAMPLES. 

1.  The  following  observations  are  made  with  the  magnet  of  a  Kew 
Magnetometer  : — 

Time  of  swing  without  brass  cylinder,  5 -05  5  sec. 

Time   with  brass  cylinder,   axis  of   cylinder  perpendicular  to  axis  of 
suspension,  8*746  sec. 
Dimensions  of  cylinder, 

length  =  9'548  cms. 
diameter  =  O998  cms. 
weight  =  63-38  gms. 
Hence  find  the  moment  of  inertia  of  the  magnet. 

2.  Determine  the  velocity  of  a  revolver  bullet  from  the  following  obser- 
vations with  the  Ballistic  Pendulum  : — 

50  gms.  placed  in  one  cylinder  caused  a  deflection  of  14°  20'. 

Distance  between  cylinder-suspensions  =  2£=80  cms. 

First  swing  =  13°  40'. 

Distance  of  line  of  fire  from  knife  edge  =  14*2  cms. 

Time  of  vibration  =  6*7  seconds. 

Mass  of  bullet  =  88  grains. 

The  revolver  used  in  this  experiment  was  a  32-calibre  Ivor  and  Johnson 
with  a  10  grain  charge  of  powder. 


CHAPTEE  XXX. 

DETERMINATION  OF  THE  VALUE  OF  GRAVITY  BY 
EATER'S   PENDULUM. 

273.  THE  acceleration  produced  by  gravity  at  any  place  is  a 
physical  constant  of  great  importance.  By  far  the  best  means  of 
finding  it  is  the  pendulum  (§  223).  But  if  a  simple  pendulum  is 
used,  no  great  accuracy  is  attainable.  The  time  of  swing  cannot 
be  exactly  found  unless  the  pendulum  can  make  many  hundred 
swings  before  the  arc,  which  must  be  small  to  begin  with,  becomes 
too  short  for  observation.  For  this  purpose  the  bob  must  be 
heavy,  and  this  requires  a  strong  wire  or  string  to  support  it. 
The  pendulum  cannot  then  be  treated  as  an  ideal  simple  pendulum, 
and  yet  it  is  not  possible  to  allow  for  the  mass  of  the  string,  or  to 
fix  the  position  of  the  centre  of  gravity  of  the  bob.  The  distance 
from  the  point  of  suspension  to  the  centre  of  the  bob  cannot  be 
measured  with  the  utmost  refinement  while  it  is  in  position,  yet 
if  it  is  taken  down,  the  string  is  no  longer  stretched  to  the  same 
length. 

Captain  Kater,  a  member  of  the  Committee  appointed  by  the 
Royal  Society  to  determine  as  accurately  as  possible  the  length  of 
the  Seconds  Pendulum  at  Greenwich,  after  many  failures  with 
the  simple  pendulum,  was  casting  about  for  some  property  of  the 
pendulum  which  would  enable  him  to  overcome  these  difficulties, 
when  he  hit  upon  a  discovery  by  Huyghens  concerning  the  Com- 
pound Pendulum  which  answered  all  requirements.  The  paper 
•describing-  his  experiments  is  in  the  Philosophical  Transactions 
for  18ia 


304 


MECHANICS 


[CHAP. 


274.  Let  C  be  the  centre  of  suspension  of  a  pendulum  of  any 
shape;  0  its  centre  of  gravity;  CO 
the  length  of  the  simple  equivalent 
pendulum.  The  point  0  was  called 
by  Huyghens  the  Centre  of  Oscilla- 
tion for  the  axis  (7;  and  he  shewed 
that  if  the  body  were  suspended 
from  a  parallel  axis  through  0,  the 
time  of  oscillation  would  be  the 
same,  and  the  length  of  the  simple 
equivalent  pendulum  would  be  the 
same ;  so  that  C  would  become  the 
centre  of  oscillation  for  the  axis  0. 
C  and  0,  the  centres  of  suspension 
and  oscillation,  are  thus  convertible, 
and  the  time  of  swing  about  each  is 
the  same  as  that  of  an  ideal  simple  pendulum  of  length  CO. 

The  proof  is  as  follows : 

Let   M  be  the  mass  of  the  pendulum,  /  its  M.I.  about  the 
centre  of  gravity  #;  and  let  CG  =  h;  C0  =  l\  OG  =  h'. 

Then 

M.  I.  of  the  pendulum  about  axis  (7  =  7  +  Mfc, 
and  M.L        „  „  „  0  =  7+  Mh'\ 

The  time  of  swing  about  0  is  that  of  the  simple  equivalent 
pendulum, 


Fig.  144. 


„    9     //+m«        /I 

i=2^v-jMr=27rv*; 


7  +  ifA2 


.'.  I=*Mhh'. 


The  time  of  swing  about  0  is 

T  -2        /I+Mh'2 
"V     Mgh' 

/MMf+l 

=  ^N--Mah> 


STT 


xxx] 


GRAVITY  BY  KATER'S  PENDULUM 


305 


275.  Drill  a  hole  through  a  board  of  any  shape.  Pass  a 
knitting  needle  through  the  hole,  and  adjust  a  simple  pendulum, 
consisting  of  a  bullet  on  a  string,  as  in  §  250,  till  the  time  of 
swing  is  the  same  for  both.  Mark  the  point  behind  the  bullet, 
and  pass  the  needle  through  a  hole  drilled  at  the  mark,  without 
altering  the  length  of  the  string.  It  will  be  found  that  the  board 
still  swings  with  the  bullet. 

Two  other  curious  properties  of  the  point  0  may  be  mentioned 
here.  Let  us  suppose  that  instead  of  the  board,  a  cricket  bat  is 
suspended  on  an  axis  C  passed  through  the  handle  where  it  is 
grasped  by  the  hands.  The  corresponding  centre  of  oscillation  0 
is  the  point  of  the  bat  with  which  a  ball  should  be  struck  so  as  to 
produce  the  greatest  effect  on  the  ball  with  a  given  swing.  It  is 
also  the  point  with  which  the  ball  must  be  struck  so  that  there 
shall  be  no  unpleasant  jar  at  the  hands.  Every  cricketer  knows 
that  when  he  makes  his  best  hits  he  does  not  feel  the  blow  of  the 
ball  at  all.  For  these  reasons  the  Centre  of  Oscillation  is  also 
called  the  Centre  of  Percussion. 

The  reason  for  these  properties  may  be  understood  without 
equations.  Imagine  the  simple  equivalent  pendulum  to  consist 
of  a  heavy  mass  suspended,  not  by  a  string,  but  by  a  rigid,  weight- 
less rod,  something  like  the  16  Ib.  hammer  on  its  handle.  Then 
for  motion  about  the  hands  the  bat  and  the  pendulum  are  dynami- 
cally similar,  and  we  may  infer  the  properties  of  the  one  from 
those  of  the  other. 


IB 

od 

Fig.  145. 


C. 


20 


306  MECHANICS  [CHAP. 

If  the  swinging  hammer  strike  an  obstacle  at  (7,  on  the  handle 
above  the  head  B,  the  momentum  of  the  head  carries  it  on  round 
C  as  a  fulcrum,  jerking  the  hands  at  A  backwards,  and  being 
partly  wasted  in  producing  the  jerk. 

If  the  obstacle  meets  the  handle  at  D,  below  the  head,  the 
momentum  of  B  jerks  A  forwards,  and  is  again  partly  wasted. 

Only  when  B  strikes  the  obstacle  directly,  as  at  E,  is  the 
whole  momentum  given  up  to  it,  as  if  the  balls  were  free  of  the 
handle.  And  then  there  is  no  jerk  at  A. 

The  same  considerations  hold  for  the  bat,  which  moves  as  if 
its  whole  mass  were  concentrated  at  the  centre  of  oscillation  or 
percussion  0.  Verify  this  by  suspending  the  board-pendulum  by 
a  light  thread  instead  of  the  knitting  needle.  Hold  the  thread 
horizontally  with  the  pendulum  at  rest  and  let  an  assistant  strike 
a  blow  on  its  edge  along  a  horizontal  line  through  the  centre  of 
percussion,  previously  determined  by  aid  of  the  bullet  and  string. 
It  will  be  found  that  quite  a  smart  blow  may  be  struck  without 
breaking  the  thread.  The  board  begins  to  turn  of  its  own  accord 
about  the  thread.  But  if  the  line  of  the  blow  be  not  exactly 
through  the  centre  of  percussion,  a  slight  tap  will  suffice  to  break 
the  thread.  Instead  of  the  board  it  is  better  to  use  a  straight  flat 
rod  suspended  by  a  hole  near  one  end.  Its  centre  of  percussion 
is  at  two-thirds  of  its  length  from  the  top. 

276.  The  property  of  the  Centre  of  Oscillation  discovered  by 
Huyghens  affords  the  means  of  constructing  the  equivalent  of  an 
Ideal  Simple  Pendulum,  and  of  measuring  with  great  exactness 
both  its  length  and  the  time  of  swing.  To  apply  it,  Kater  con- 
structed a  pendulum  of  a  bar  of  plate  brass,  1J  in.  wide  and 
J  in.  thick.  Two  knife  edges  of  hard  steel  were  ground  true  and 
firmly  fixed  near  the  ends  at  a  distance  of  about  39*4  inches 
apart.  A  brass  weight  of  2  Ibs.  7  oz.  was  fixed  between  one  end 
of  the  bar  and  the  nearer  knife  edge ;  a  second  weight  of  7  J  oz. 
was  made  to  slide  on  the  bar  near  the  other  knife  edge  and 
between  them  ;  and  a  small  slider  of  4  oz.,  capable  of  fine  adjust- 
ment by  a  screw,  was  placed  near  the  middle  of  the  bar. 

The  pendulum  could  be  suspended  by  either  knife  edge  resting 


XXX]  GRAVITY   BY   EATER'S   PENDULUM  307 

upon  polished  agate  planes  let  into  a  firm  support.  The 
times  of  vibration  about  the  two  knife  edges  were  roughly 
found ;  the  second  weight  of  7£  oz.  was  moved  along  the 
bar  till  they  were  nearly  equal ;  and  then  the  slider  was 
carefully  adjusted  till  the  time  of  vibration  about  either 
knife  edge  was  practically  the  same.  This  would  therefore 
be  the  time  of  vibration  of  an  ideal  pendulum  of  a  length 
equal  to  the  distance  between  the  knife  edges.  It  re- 
mained to  find  this  distance  and  the  time  of  swing 
accurately. 

277.  The  pendulum  was  mounted  on  a  comparator 
Measurement  of  under  reading  microscopes  which  were 
the  Length.  brought  over  the  knife  edges,  and  com- 

pared with  standard  scales.  The  mean  results  of  three 
sets  of  measurements  taken  at  different  times  were  within 
one  ten-thousandth  of  an  inch  of  each  other.  The  cor- 
rected mean  of  all  the  sets  was 

39-44085  inches  at  62°  F. 


278.     The  pendulum  was  compared   with   an   astro- 
nomical  clock   beating  seconds,  and  the 

Determination  of 

the  Time  of  rate  of  the  clock  was  found  from  transit  -p^  145. 

observations  taken  every  day  during  the 
experiments,  and  often  more  frequently. 

It  is  not  possible  to  observe  the  exact  time  of  a  chosen  number 
of  vibrations,  for  this  would  require  the  fraction  of  a  second  to  be 
estimated  either  by  the  eye  or  ear.  Nor  could  the  number  of 
vibrations  in  a  fixed  interval  be  observed,  for  this  would  require  a 
fraction  of  a  vibration  to  be  estimated.  The  device  by  which  this 
difficulty  was  avoided  is  known  as  the  Method  of  Coincidences. 
The  pendulum  was  mounted  on  a  firm  support  in  front  of  the 
clock.  (The  second  clock  in  the  figure  was  only  used  as  a  check.) 

A  small  disc  of  white  paper  was  fixed  on  the  bob  of  the  clock 
pendulum,  of  the  same  width  as  the  two  slips  of  blackened  deal 
projecting  from  the  ends  of  the  Kater  pendulum.  At  a  distance 
of  nine  feet  in  front  of  the  clock  was  a  telescope  with  the  field  of 

20—2 


308 


MECHANICS 


[CHAP. 


view  limited  by  a  diaphragm  to  a  narrow  vertical  slit  exactly  wide 
enough  to  shew  one  of  the  deal  slips  or  the  paper  disc.  If  both 
pendulums  were  at  rest,  the  disc  would  be  just  hidden  by  the 
deal  slip. 

The  distance  between  the  knife  edges  had  been  fixed  at  about 
39'4  in.  so  that  the  period  of  vibration  should  be  very  nearly  that 


ig.  147. 


XXX]  GRAVITY  BY   RATER'S   PENDULUM  309 

of  the  clock  pendulum,  but  a  little  greater.  "  If  both  pendulums 
be  now  set  in  motion,  the  brass  pendulum  a  little  preceding  that 
of  the  clock,  the  slip  of  deal  will  first  pass  through  the  field  of 
view  of  the  telescope  at  each  vibration  and  will  be  followed  by 
the  white  disc.  But  the  clock  will  gain  on  the  pendulum,  so  that 
the  white  disc  will  gradually  approach  the  slip  of  deal,  and  at  a 
certain  vibration  will  be  wholly  concealed  by  it.  The  minute  and 
second  at  which  this  total  disappearance  is  observed  must  be 
noted.  The  pendulums  will  now  be  seen  to  separate,  and  after  a 
time  will  again  approach  each  other,  when  the  same  phenomenon 
will  take  place.  The  interval  between  the  two  coincidences  in 
seconds  will  give  the  number  of  vibrations  made  by  the  clock; 
and  the  brass  pendulum  must  have  made  two  less  than  the  clock. 
Hence  by  simple  proportion  the  time  of  vibration  of  the  pendulum 
is  known."  Kater  calculated  the  number  of  vibrations  made  in 
24  hours,  or  86,400  seconds. 

To  shew  the  accuracy  attainable  by  this  method,  let  us  suppose 
an  error  of  one  whole  second  in  observing  a  coincidence  (it  will 
be  noticed  that  no  fractions  have  to  be  estimated).  The  interval 
between  coincidences  in  Rater's  experiments  was  about  530  seconds. 
With  this  value  the  pendulum  must  have  made  528  swings,  so  that 
its  time  of  swing  would  be  530/528  seconds,  i.e.  T003787  seconds. 
If  by  error  the  interval  were  taken  as  531  seconds,  the  time 
would  be  531/529  =  T0037807  seconds.  This  does  not  differ  from 
the  true  value  by  one  part  in  100,000. 

279.     The  observed  result  must  be  corrected  for 

(1)  Expansion  or  contraction  of  the  pendulum;   due  to 
difference  of  temperature  at  the  time  of  the  experiment  from  the 
temperature,  62°  F.,  at  which  the  length  between  knife  edges  was 
measured. 

(2)  Variation  from  Simple  Harmonic  Motion  due  to  the 
size  of  the  arc.     The  arc  was  read  by  a  whalebone  pointer  moving 
over  a  scale  of  degrees.     This  correction  is  called  the  "  reduction 
to  infinitely  small  arcs." 

(3)  Buoyancy  of  the  air,  making  the  weight  different  from 
what  it  would  be  in  vacuo.     This  depended  on  the  height  of  the 
barometer. 


310 


MECHANICS 


[CHAP. 


280.  Twelve  sets  of  observations  similar  to  Table  I.  were 
made.  The  results  are  summarized  in  Tables  II.  and  III.  In 
Table  II.  it  will  be  observed  how  nearly  the  time  of  swing  was  the 
same  for  the  two  positions  of  the  pendulum. 


TABLE  I 


1 

Slider   19  divisions                                                                                                                 -RnrrvmPtPr 

Clock  gaining  0",18                 GREAT  WEIGHT  above                                    2<H)0 
on  mean  time                                                                                                     ' 

Temp. 

Time  of  co- 
incidence 

Arc  of 
vibration 

Mean 
arc 

Jnterv.in 
seconds 

No.  of 
vibrats. 

Vibrations 
in  24  hours 

Corr. 
for  arc 

Vibrations 
in  24  hours 

July 
3rd 

M 

68,3 

68,4 

m,  s. 
45.  3 
53.27 
1.51 
10.16 
18.42 

o 

1,23 
1,03 
0,87 
0,74 
0,63 

0 

1,13 

0,95 
0,80 
0,68 

504 
504 
505 
506 

502 
502 
503 
504 

86057,16 
86057,16 

86057,82 
86058,49 

S. 

2,08 
1,47 
1,04 
0,75 

Mean 
Clock 

86059,24 
86058,63 
86058,86 
86059,24 

86058,99 
+  0,18 

68,4 

mean 

86059,17 

GREAT  WEIGHT  below 
i 

68,4 
68,5 

24.31 
32.54 
41.18 

49.42 
58.  8 

1,24 
1,11 

0,99 
0,90 

0,82 

1,17 
1,05 
0,94 
0,86 

503 
504 
504 
506 

501 
502 
502 
504 

86056,47 
86057,16 
86057,16 
86058,49 

2,23 
1,80 
1,44 
1,20 

Mean 
Clock 
Temp 

86058,70 
86058,96 
86058,60 
86059,69 

86058,99 
+  0,18 
+  0,04 

68,5 

mean                                                                                             86059,21 

The  results  of  such  of  the  preceding  experiments  as  are  to 
be  used  for  calculating  the  length  of  the  seconds  pendulum,  are 
brought  under  one  view  in  the  following  table : 


xxx] 


GRAVITY  BY  KATER'S  PENDULUM 


311 


TABLE  II. 


Place  of 
the  slider 

Expt. 

Temp. 

Barom. 

No.  of  vibrations. 
Great  wt.  above 

Diff. 

No.  of  vibrations. 
Great  wt.  below 

Vibs.  in 
excess 
or  defect 

23 

A 

68,7 

29,76 

86059,39 

,03 

86059,42 

23 

B 

71,3 

29,86 

86057,70 

,23 

86057,93 

_ 

23 

C 

71,4 

29,86 

86057,93 

,23 

86057,70 

+ 

23 

D 

73,1 

29,95 

86056,54 

,43 

86056,97 

— 

Pendulum  re-measured 

21 

E 

69,3 

29,70 

86058,88 

,06 

86058,94 

20 

F 

69,3 

29,70 

86058,89 

,12 

86059,01 

_ 

20 

G 

68,5 

29,70 

86059,03 

,19 

86059,22 

_ 

18 

H 

68,7 

29,70 

86059,36 

,11 

86059,25 

+ 

18 

I 

69,3 

29,70 

86059,19 

,16 

86058,93 

+ 

18 

K 

69,3 

29,70 

86059,14 

,31 

86058,83 

+ 

19 

L 

68,1 

29,90 

86059,26 

,04 

86059,22 

+ 

19 

M 

68,4 

29,90 

86059,17 

,04 

86059,21 

Mean 

86058,71 

86058,72 

"Using  the  vibrations  when  the  great  weight  was  below,  as 
being  nearer  to  the  truth  than  in  the  other  position  of  the 
pendulum,  we  obtain  the  following  results." 

TABLE  III. 


Expt. 

Temp. 

Barom. 

Vibrations 
in  24  hours 

Length  of  the 
seconds  pen. 
in  air 

Corr.  for 
the 
atmosphere 

Length  of  the 
seconds  pend. 
in  vacuo 

Difference 
from  the 
mean 

A 

68,7 

29,76 

86059,42 

39,13313 

,00544 

39,13857 

+  ,00028 

B 

71,3 

29,86 

86059,93 

39,13278 

,00544 

39,13822 

-  ,00007 

C 

71,4 

29,86 

86057,70 

39,13260 

,00544 

39,13804 

-  ,00025 

D 

73,1 

29,95 

86056,97 

39,13259 

,00544 

39,13803 

-  ,00026 

E 

69,3 

29,70 

86058,94 

39,13293 

,00544 

39,13837 

+  ,00008 

F 

69,3 

29,70 

86059,01 

39,13298 

,00544 

39,13842 

+  ,00013 

G 

68,5 

29,70 

86059,22 

39,13286 

,00545 

39,13831 

+  ,00002 

H 

68,7 

29,70 

86059,25 

39,13296 

,00544 

39,13840 

+  ,00011 

I 

69,3 

29,70 

86058,93 

39,13291 

,00544 

39,13834 

+  ,00005 

K 

69,3 

29,70 

86058,83 

39,13282 

,00544 

39,13825 

-  ,00003 

L 

68,1 

29,90 

86059,22 

39,13271 

,00548 

39,13819 

-  ,00009 

M 

68,4 

29,90 

86059,21 

39,13281 

,00548 

39,13829 

-  ,00000 

Mean 

39,13829 

312  MECHANICS  [CHAP.  XXX 

The  mean  value  was  then  corrected  for  the  height  of  Portland 
Place,  where  the  observations  were  made,  above  the  sea  level. 
The  final  result  was 

39-1386  inches 

for  the  length  of  the  pendulum  vibrating  seconds  at  the  level  of 
the  sea  in  the  latitude  of  London. 


EXAMPLE. 

From  Eater's  value  of  the  length  of  the  seconds  pendulum  determine 
the  value  of  gravity  at  the  sea-level  in  London. 


CHAPTER   XXXI. 

THE  CONSTANT  OF  GRAVITATION,   OR  WEIGHING  THE 
EARTH.     THE   CAVENDISH  EXPERIMENT. 

281.  JUST  twenty  years  before  Kater  determined  accurately 
the  value  of  the  acceleration  of  gravity  at  the  earth's  surface, 
Henry  Cavendish  succeeded  in  measuring  another  important 
physical  constant,  which  was  necessary  to  complete  the  statement 
of  the  law  of  universal  gravitation. 

According  to  Newton's  Law,  the  force  of  attraction  between 
any  two  masses  is  directly  proportional  to  the  product  of  the 
masses  and  inversely  proportional  to  the  square  of  their  distance. 
This  enables  us  to  compare  attractions,  but  before  we  can  calculate 
the  actual  value  of  the  attraction  between  two  given  masses  in 
units  of  force  otherwise  known  to  us,  we  must,  as  in  every  case  of 
proportion,  know  what  it  is  in  some  standard  instance. 

Suppose,  for  example,  we  agree  to  measure  masses,  distances, 
and  forces  in  grammes,  centimetres,  and  dynes,  and  know  that  the 
attraction  exerted  by  a  particle  of  mass  one  gramme  on  an  equal 
particle  distant  one  centimetre  from  it  is  G  dynes.     Then  : 
Attraction  of  1  gm.  on  1  gm.  at  1  cm.  is  0        dynes 

„   mx      „       1       „        1      „        Gml 
„  „   raj      „      ma     „        1      „ 


G  is  called  the  constant  of  Gravitation.  Its  value,  like  that  of 
the  dyne  and  the  poundal,  can  only  be  found  by  an  experiment. 
If  we  chose  pounds,  feet,  and  poundals  for  our  units,  G  would  of 
course  have  a  different  numerical  value. 


314  MECHANICS  [CHAP. 

282.  From  another  point  of  view  the  determination  of  G  is 
equivalent  to  finding  the  mass  of  the  earth  in  terms  of  the 
ordinary  units  of  mass. 

Newton  had  been  able  to  compare  the  mass  of  the  Earth,  or  of 
any  planet  which  had  a  satellite,  with  that  of  the  Sun.  For  let  S, 
E,  M  be  the  masses  of  the  Sun,  Earth,  and  Moon;  let  R,r  be  the 
radii  of  the  Earth's  and  Moon's  orbits,  supposed  circular ;  and  let 
Tet  Tm  be  the  times  of  revolution  of  the  Earth  and  Moon  in  their 

orbits.     Then  (§77), 

4-7T2 
acceleration  of  Earth  to  Sun  =  -™-  .  R, 

A       2 

and  acceleration  of  Moon  to  Earth  =  ~— .  r. 

But,   by  the  law  of  gravitation,  force  ex-  _  ~   3.  E 
erted  by  Sun  on  Earth  ~W  ' 

C1    Sf    T? 
Therefore,  by  Second  Law  of  Motion,  ac-         '     '  ~ 

celeration  of  Earth  to  Sun  = — —  =  G .  ^ . 

£T 

Similarly,  acceleration  of  Moon  to  Earth  =  G  .  — . 


Therefore 

4?r2          n  E 
and  jrvfa&.p; 

E      r3    T? 
whence  8  =  ^'T^' 

in  which  R,  r,  Te,  Tm  are  known  from  observation. 

But  this  does  not  tell  us  what  the  mass  of  the  Earth  or  the 
Sun  is  in  pounds  or  grammes. 

If,  however,  we  knew  the  value  of  G  in  dynes,  then,  since  the 
weight  of  a  gramme  is  the  attraction  exerted  on  it  by  the  whole 
mass  of  the  Earth,  supposed  collected  at  its  centre  (§  238),  we 
should  have 

V    1 

G  .  — '—  =  weight  of  1  gm.  at  the  surface  of  the  Earth 

=  981  dynes ; 


XXXI]  THE   CAVENDISH   EXPERIMENT  315 

QQ-J 

whence  -Z?=-~— .a2, 

Or 

and  E  is  given  in  grammes  when  a,  the  radius,  is  expressed  in 
centimetres. 

The  Earth  would  thus  be  "  weighed,"  i.e.  expressed  in  terms  of 
a  known  unit  of  mass.  Or,  we  could  express  A,  the  mean  density 
of  the  Earth,  compared  with  water,  since 

E  =  ±7ra3 .  A  gms., 
so  that  A  is  known,  if  E  is  known. 

283.  The  problem  is  thus  reduced  to  finding  by  an  experiment 
the  actual  attraction  in  dynes  between  two  known  masses. 

Now  the  attraction  of  the  whole  earth  on  one  gramme  only 
amounts  to  the  weight  of  one  gramme.  Thus  the  attraction 
between  any  two  masses  accessible  to  us  must  be  a  very  minute 
force. 

Two  lines  of  attack  are  open  to  us.  Either  we  must  choose 
for  one  of  the  masses  some  large  natural  mass,  such  as  a  mountain, 
in  the  hope  of  increasing  the  force  up  to  a  measurable  magnitude ; 
or  else  apparatus  of  extreme  sensitiveness  must  be  devised,  if  we 
are  to  detect  the  attraction  between  such  masses  as  can  be  dealt 
with  in  a  laboratory. 

284.  The  first  method  was  tried  by  Bouguer  in   1740,  on 
Chimborazo,  a  mountain  20,000  feet  high  in  the  Andes ;  and  again, 
with  great  care,  by  Maskelyne  in  1772,  on  Schehallion  in  Scotland. 
Each  of  these  observers  suspended  a  plumb  line  on  the  side  of  the 
mountain,  and  noted  the  deflection  from  the  true  vertical,  which 
could  be  determined  astronomically.     Similar  experiments  have 
since  been  carried  out  at  Arthur's  Seat  near  Edinburgh. 

In  1854  Airy,  another  English  Astronomer  Royal,  observed 
the  change  in  the  time  of  swing  of  a  pendulum  when  it  was 
removed  from  the  surface  of  the  earth  to  the  bottom  of  the  Harton 
coal  pit,  1250  feet  deep.  Part  of  this  was  to  be  attributed  to 
change  of  distance  from  the  centre,  and  could  be  calculated.  The 
rest  was  due  to  the  removal  of  a  layer  of  the  earth's  crust 
1250  feet  thick,  since  the  outer  shell  has  no  attraction  on  the 


316 


MECHANICS 


[CHAP. 


pendulum  (§  237).  Others  (Carlini  in  1821  on  Mont  Cenis,  and 
Mendenhall  in  1880  on  Fujiyama  in  Japan)  made  similar  de- 
terminations by  removing  the  pendulum  to  the  top  of  a  high 
mountain. 

The  objection  to  all  these  methods  is  that  the  exact  shape, 
and  above  all  the  density  of  the  rocks  and  strata  in  the  disturbing 
mass  cannot  be  accurately  determined. 

285.  The  first  to  plan  an  experiment  on  the  laboratory  method 
was  the  Rev.  John  Michell,  who  constructed  the  apparatus,  but 
did  not  live  to  make  the  experiment.  His  apparatus  passed  to 
Professor  Wollaston  of  Cambridge,  who  handed  it  over  to  Henry 
Cavendish,  noted  for  his  skill  as  an  experimenter  in  electricity  arid 
other  branches  of  Physics. 

In  carrying  out  the  famous  experiment  which  goes  by  his 
name,  Cavendish  retained  MichelFs  original  idea,  and  even  the 
dimensions  of  the  apparatus ;  but  found  it  advisable  to  reconstruct 
most  of  it.  The  account  of  his  work  was  communicated  to  the 
Royal  Society  in  1798,  and  is  to  be  found  in  the  volume  of  the 
Philosophical  Transactions  for  that  year. 


Fig.  148. 


XXXI]  THE   CAVENDISH    EXPEKIMENT  317 

286.  The  apparatus  was  essentially  a  Torsion  Pendulum, 
similar  to  that  of  §  264.  About  the  same  time  Coulomb  in- 
dependently applied  this  instrument  to  the  measurement  of  small 
electric  and  magnetic  attractions. 

It  consisted  of  a  slender  deal  rod  hh,  6  feet  long,  stiffened  by 
an  upright  gm  braced  with  a  silver  wire  hgh,  and  suspended  by 
a  fine  silvered  copper  wire  39  J  inches  long.  The  wire  was 
fastened  by  clamps  to  the  upright  on  the  rod  and  to  the  torsion 
head  FF.  The  balls  to  be  attracted  were  two  leaden  spheres  xx, 
each  about  two  inches  in  diameter,  hung  from  the  ends  of  the  rod. 
The  whole  was  enclosed  in  a  narrow  wooden  case  with  glass 
windows,  and  the  case  was  supported  by  levelling  screws  SS  on 
firm  uprights  let  into  the  ground.  By  means  of  an  endless  screw 
worked  by  the  rod  K  the  torsion  head  FF  could  be  turned  from 
outside  so  as  to  make  the  rod  lie  centrally  in  the  case. 

The  attracting  masses  were  leaden  spheres  WW,  about  8  inches 
in  diameter,  supported  by  copper  rods  from  a  wooden  cross  rod  rr. 
This  rod  could  be  turned,  also  from  outside,  by  means  of  the  rope 
and  pulley  mMM,  so  as  to  bring  the  weights  up  to  within  1/5  inch 
from  the  case  opposite  the  suspended  balls,  where  they  were 
stopped  by  blocks  of  wood  attached  to  the  walls  of  the  room, 
or  swing  them  round  to  the  corresponding  position  on  the  opposite 
side.  It  is  obvious  that  in  the  two  positions  the  attractions  of  the 
large  weights  on  the  suspended  balls  would  tend  to  twist  the 
pendulum  from  its  central  position  in  opposite  directions. 

Cavendish  calculated  that  the  attractions  to  be  observed  could 
not  be  greater  than  the  l/50,000,000th  part  of  the  weight  of  the 
balls,  so  that  a  very  minute  disturbing  force  would  destroy  the 
success  of  the  experiment;  and  rightly  judging  that  the  most 
difficult  disturbance  to  guard  against  would  be  that  due  to 
wandering  currents  of  air  inside  the  case  caused  by  unequal 
heating,  he  "resolved  to  place  the  apparatus  in  a  room  which 
should  remain  constantly  shut,  and  to  observe  the  motion  of  the 
arm  from  without  by  means  of  a  telescope."  Hence  the  arrange- 
ments for  moving  the  weights  and  the  torsion  head  from  outside. 

To  observe  the  deflections  a  small  ivory  scale,  divided  to 
twentieths  of  an  inch,  was  set  up  near  each  end  of  the  rod ;  and 
the  rod  carried  other  ivory  scales  which  served  as  verniers  capable 


318  MECHANICS  [CHAP. 

of  reading  the  fixed  scales  to  one-fifth  of  a  division :  so  that  he 
could  observe  to  one-hundredth  of  an  inch  and  estimate  even  more 
closely.  The  lamps  for  illuminating  the  scales  and  the  telescopes 
for  reading  them  were  both  placed  outside.  No  other  light  was 
admitted  to  the  room. 


287.     (1)   Steady  deflection. 

Let  M,  m  be  the  masses  of  the  large  and  small  spheres  ;  d  the 

Theory  of  the      distance  between  their  centres  when  the  rod  is 

Experiment.        central  ;  2a  the  length  of  the  deal  rod.    Let  r  be 

the  moment  of  the  couple  required  to  deflect  the  rod  through  one 

radian;  0  the  circular  measure  of  the  angle  through  which  it  is 

deflected  when  the  weights  are  moved  up  to  one  side. 


Then  0.r  =  2G..a   ......  .  .................  (1). 

(2)     Time  of  vibration. 

Let  /  be  the  moment  of  inertia  of  the  rod  and  suspended  balls 
about  the  axis  ;  and  let  T  be  the  time  of  vibration. 

Then  T=27r>/-  ..............................  (2). 

Eliminating  T  between  (1)  and  (2)  we  have 
47T2/  Mma 


Therefore  faZg^JL-;, 

T2      a    Mm 

The  quantities  M,  m,  d,  a  are  known,  and  I  can  be  calculated. 
It  only  remains  to  observe  6  and  T. 

288.  The  following  Table  contains  a  typical  set  of  observations 
taken  from  Cavendish's  paper. 

289.  The  words  "  negative  "  and  "  positive  "  distinguish  those 
Explanation  of      positions  of  the  attracting  weights  in  which  they 

tended  to  make  the  rod  rest  at  the  lower  and 
higher  numbers  on  the  scale  respectively. 


XXXI] 


THE   CAVENDISH   EXPERIMENT 


319 


EXPERIMENT  XIV.     May  26. 

Weights  in  negative  position. 


Extreme 
points 

Divisions 

Time 

Point  of  rest 

Time  of  mid.  of 
vibration 

h.      ' 

,/ 

h. 

16-1 

9       18 

0 

16-1 

24 

0 

16-1 

46 

0 

16-1 

49 

0 

16-1 

Weights  moved  to  positive  position. 


27-7 

23 
22 

10        0 

1 

46) 
16) 

— 

10       1       1 

17-3 



__ 

22-37 

22 

7 

68] 

8        if 

23 

8 

27  J 

o 

27-2 

— 

— 

22-5 

23 
22 

15 

32  j 

— 

15       9 

18-3 

— 



— 

22-65 

26-8 







22-75 

19-1 

— 



— 

22-85 

26-4 







22-97 

23 
22 

43 
44 

40) 

22J 

— 

43      32 

20 

— 



23-15 

26-2 

22 
23 

49 
50 

53) 

37  J 

— 

50      41 

Weights  moved 

to  negative  position. 

12-4 

16 
17 

11        7 
8 

53) 

27} 

— 

11        8      25 

21-5 

— 

— 

17-02 

17 
16 

15 
16 

30) 

3J 

— 

15      27 

12-7 

— 



16-9 

20-7 







16-85 

13-3 

— 

— 

— 

16-82 

20 

— 





16-72 

13-6 

— 

— 



16-67 

16 

17 

11      50 
51 

33) 
18  J 

— 

11      50      58 

19-5 

— 

— 

16-65 

17 
16 

57 
58 

53) 

44J 

— 

58        6 

14 

Motion  of  arm  by  moving  weights  from  -  to  +  =  6-27. 

„     +  to -=6-13. 
Time  of  vibration  at  +          =7'  6" 

=  7'  6" 


320  MECHANICS  [CHAP. 

The  first  column  gives  the  successive  turning  points  as  the  rod 
vibrated  over  the  divisions  of  the  scale.  From  every  three  suc- 
cessive points  the  division  at  which  the  rod  would  ultimately  come 
to  rest  was  determined  as  in  the  method  of  weighing  by  oscillations 
(§  29).  The  results  are  given  in  column  4. 


Thus  ±(  +  17-3  1  =  22-37. 


Columns  2  and  3  give  the  times  of  passing  the  two  divisions 
of  the  scale  (23  and  22)  between  which  the  point  of  rest  lay ;  and 
from  these  is  calculated  the  moment  at  which  the  rod  must  have 
passed  the  point  of  rest,  22'37.  This  is  recorded  in  column  5,  from 
which  T,  the  time  of  vibration,  is  found  to  be  7'  6". 

The  ivory  scale  was  38*3  inches  from  the  centre  of  motion,  and 
was  divided  to  twentieths  of  an  inch.  The  circular  measure  of 
the  angle  turned  through  by  the  rod  when  the  weights  were 
shifted  from  the  negative  to  the  positive  position  was  therefore 

j  (6-27  + 613)  x?k=  31 
38-3  ~  766 ' 

The  angle  6  was  half  this. 

290.  It  appears  from  column  4  that  after  the  weights  have 
been  moved  to  the  positive  position  the  point  of  rest  steadily 
shifts  through  about  a  division  in  an  hour  towards  the  upper  end 
of  the  scale ;  and  the  reverse  occurs  after  they  have  been  moved 
back  again.  This  "  creeping  "  occurred  in  all  the  experiments,  and 
Cavendish  felt  that,  as  it  might  indicate  a  possible  source  of  error, 
it  had  to  be  explained.  He  tracked  it  to  its  source  with  great 
ingenuity. 

His  first  idea  was  that  the  wire  had  possibly  been  twisted 
slightly  beyond  its  elastic  limit,  and  might  gradually  take  a  set, 
from  which  it  partially  recovered  on  reversing  the  position  of  the 
weights.  Accordingly  the  large  weights  were  moved  to  the 
midway  position,  and  the  torsion  head  turned  enough  to  press 
the  suspended  balls  against  the  sides  of  the  case,  and  twist  the 
wire  15  divisions  more.  But  though  they  were  left  thus  for  two 
or  three  hours,  the  rod  returned  to  its  natural  position,  when  the 
torsion  head  was  turned  back,  and  shewed  no  tendency  to  creep. 


XXXl]  THE   CAVENDISH    EXPERIMENT  321 

Next,  he  suspected  that  the  leaden  weights  and  balls  might 
contain  traces  of  iron  or  other  magnetic  impurity.  In  that  case, 
as  the  rod  happened  to  be  set  up  east  and  west,  they  would 
gradually  become  feebly  magnetized  by  the  earth's  field  in  the 
direction  of  the  line  joining  their  centres,  and  thus  attract  each 
other  magnetically.  To  test  this  he  made  an  arrangement  by 
which  he  could  rotate  the  large  weights  on  the  copper  rods  through 
half  a  turn,  without  entering  the  room.  The  weights  were  moved 
up  to  the  case  overnight,  and  the  pendulum  allowed  to  come  to 
rest.  In  the  morning  they  were  turned  halfway  round  on  the 
copper  rods,  so  as  to  reverse  their  magnetic  poles.  But  he  could 
not  detect  any  effect  upon  the  pendulum.  The  large  weights  were 
then  replaced  by  ten-inch  magnets  which  could  be  turned  end  for 
end.  This  also  failed  to  affect  the  pendulum.  So  that  there  was 
no  trace  of  magnetic  effect  either  in  the  large  weights  or  in  the 
suspended  balls. 

Finally,  he  concluded  that  the  effect  must  be  due  to  a  difference 
of  temperature  between  the  weights  and  the  air  in  the  case.  If 
the  weights  were  warmer,  they  would,  for  some  time  after  being 
moved  up,  go  on  warming  the  air  in  the  case,  thus  causing  an 
upward  current  between  the  near  side  and  the  suspended  balls, 
tending  to  draw  the  balls  towards  the  walls  of  the  case.  This  was 
tested  by  setting  lamps  beneath  the  weights  in  the  midway 
position,  and  leaving  them  overnight  to  burn  out.  On  moving  the 
weights  up  to  the  case  in  the  morning  a  much  larger  creeping 
effect  was  obtained.  If,  on  the  contrary,  the  weights  were  cooled 
by  leaving  pieces  of  ice  to  melt  on  them,  a  large  effect  was 
obtained,  but  in  the  opposite  direction.  Holes  were  then  drilled 
in  the  weights  and  small  thermometers  inserted,  other  thermo- 
meters being  hung  against  the  case,  in  such  a  way  that  they  could 
all  be  read  by  the  observing  telescopes.  It  then  appeared  that 
there  was  always  a  difference  of  temperature  amounting  to  one  or 
two  degrees,  between  the  weights  and  the  air,  the  weights  being 
the  warmer.  The  effect  was  thus  accounted  for  satisfactorily. 

291.  In  computing  the  final  results  small  corrections  had  to 
be  applied  as  follows:  (1)  for  the  attraction  of  the  weights  on  the 
rod;  (2)  for  the  attraction  of  the  weights  on  the  farther  ball; 

f.  21 


322 


MECHANICS 


[CHAP. 


(3)  for  the  attraction  of  the  copper  rods  on  the  balls  and  arm ; 

(4)  for  the  attraction  of  the  case  on  the  balls  and  arm ;  (5)  for  the 
alteration  of  the  attraction  of  the  weights  on  the  balls,  according 
to  the  position  of  the  arm,  and  for  the  effect  which  that  has  on  the 
time  of  vibration.     "  None  of  these  corrections,  indeed,  except  the 
last,  are  of  much  signification,  but  they  ought  not  entirely  to  be 
neglected."     In  fact  they  were  most  carefully  worked  out. 

From  twenty-nine  sets  of  observations  values  of  the  density  of 
the  earth  were  obtained  ranging  from  4*88  to  579,  with  a  mean 
value  of  5-448. 

292.  Since  the  time  of  Cavendish  the  density  of  the  earth  has 
been  determined  by  many  observers,  some  of  whom  employed 
modifications  of  his  method,  while  others  used  some  form  of 
balance,  i.e.  a  pendulum  oscillating  in  a  vertical  plane,  the 
attracting  masses  being  placed  beneath  it.  The  results  obtained 
may  be  summarized  as  follows : 


Date 

Observer 

Value  of  A 

1798 

Cavendish 

5-448 

1837 

Keich 

5-49 

1841-2 

Baily 

5-674 

1849 

Eeich 

5-58 

1870 

Cornu  and  Bailie 

5-5 

1878 

Poynting 

5-4934 

1878-81 

Von  Jolly 

5-69 

1886 

Wilsing 

5-579 

1895 

Boys 

5-5270 

1896 

Braun 

5-52725 

1898 

Eicharz  and  Krigar-Menzel 

5-505 

293.  The  student  should  take  an  opportunity  of  referring  to 
Cavendish's  original  paper,  for  it  would  not  be  easy  to  find  a  more 
superb  example  of  the  triumph  of  patience  and  accuracy,  combined 
with  delicate  manipulation,  over  experimental  difficulties  that 
might  seem  well-nigh  insuperable.  There  is  even  a  lesson  to  be 
learned  from  the  curious  irony  of  fate  by  which,  after  all  his  care, 
Cavendish  gave  the  mean  value  of  his  results  as  5'48,  instead  of 
5'448,  through  a  mere  arithmetical  slip  in  finding  the  average! 
Very  often  in  the  course  of  physical  measurements  the  attention  is 
so  absorbed  in  overcoming  experimental  difficulties,  or  recording 


XXXI]  THE   CAVENDISH    EXPERIMENT  823 

fractions  of  a  millimetre  or  a  second,  that  far  more  important  con- 
siderations are  overlooked,  or  a  mistake  is  made  in  the  number  of 
centimetres  or  whole  degrees.  It  is  not  safe  in  laboratory  work  to 
hold  by  the  maxim  about  taking  care  of  the  pence.  The  pounds 
must  have  at  least  an  equal  share  of  attention. 

By  way  of  pendant  to  this  proof  that  men  of  genius  are  not 
infallible,  we  may  cite  Newton's  celebrated  estimate  of  the  density 
of  the  earth  to  shew  that  their  intuitions  are  sometimes  lucky. 
With  little  to  go  upon  except  the  argument  that  if  the  earth  were 
lighter  than  water,  it  would  emerge  from  the  ocean  like  a  cork  on 
one  side  or  the  other,  Newton  proceeds  (Principia,  Book  ill. 
Prop.  10),  "  Since,  therefore,  the  common  matter  of  our  earth  on 
the  surface  thereof,  is  about  twice  as  heavy  as  water,  and  a  little 
lower,  in  mines,  is  found  about  three  or  four,  or  even  five  times 
more  heavy ;  it  is  probable  that  the  quantity  of  the  whole  matter 
of  the  earth  may  be  five  or  six  times  greater  than  if  it  consisted 
all  of  water,  especially  as  I  have  before  shewed  that  the  earth  is 
about  four  times  more  dense  than  Jupiter/' 

The  true  value  is  almost  exactly  between  the  limits  thus 
assigned !  And  this  is  only  one  of  many  guesses  thrown  out  by 
that  commanding  genius  in  all  branches  of  Physics  that  have  since 
been  shewn  to  be  near  the  truth  by  the  course  of  modern  science. 


EXAMPLES. 

1.  With  the  following  values  : 

Time  of  revolution  of  earth  about  sun = 365  d.  6h.     9m. 
„  moon     „     earth  =  27  d.  7h.  43m. 

Mean  radius  of  earth's  orbit  =  1  '487  x  1013  cms. 
„  moon's    „     =3'84   x!010cms. 

shew  that  the  mass  of  the  sun  is  about  324,800  times  that  of  the  earth. 

2.  Taking  Cavendish's  value  for  the  density  of  the  earth,  A  =  5'448,  and 
assuming    the    earth's    mean    radius    to    be    6'37  x  108   cms.,    shew    that 
G  =  6-748xlO-8  dynes. 

3.  Boys  found  for  G  the  value  6'6576xlO"8  dynes.     Shew  that  this 
makes  A  =  5'52 

4.  Shew  that  the  mass  of  the  earth  is  about  6  x  10-1  English  tons. 


ANSWERS   TO   THE  EXAMPLES. 

(The  numbers  in  the  Examples  have  not,  for  the  most  part,  been  chosen 
to  give  exact  answers,  but  are  either  taken  from  actual  observations,  or  such 
as  would  occur  in  practical  work.  The  answers  should  be  worked  out 
numerically  to  an  accuracy  of  about  one  per  cent.,  by  aid  of  four-place 
mathematical  tables.  The  student  will  find  the  practice  in  calculation  quite 
worth  the  trouble.  Such  forms  as  jrA/2,  e27r,  &c.  should  not  be  left  in  an 
answer,  except  for  a  special  reason,  but  should  be  evaluated  by  the  Tables. 

Unless  the  contrary  is  stated,  g  may  be  taken  =  32,  and  tons  are  English 
tons  of  2240  Ibs.) 

CHAPTER  II.,  p.  21. 

1.     100  Ibs.  2.     7-854  in.  3.     270  Ibs.  4.     2880  Ibs. 

5.  240  Ibs.  6.     60  Ibs.  7.     (1)  14  Ibs.,  (2)  21 J  Ibs. 

CHAPTER  III.,  p.  30. 

3.    3  ft.  4  in.  from  weight  2.  4.     a  V3.  5.    ^  in. 

6.  2f  in.  from  centre.  7.     2685-2  inches  from  centre  of  earth. 
9.     3f  in.  from  base.                  10.     16  Ibs. ;  8  feet. 

CHAPTER  IV.,  p.  39. 

1.     The  arms  are  unequal.     20-599  gms.  2.     38 '5  mgms. 

4.     11  Ibs.  10 18!  oz.  6.     ^.  7.     -058  in. 

CHAPTER  V.,  p.  47. 
1.     8800  Ibs.  2.     390  Ibs.  3.     5  :  \/41. 

CHAPTER  VII.,  p.  63. 
1.     3-73  Ibs.  2.     —  cot  a  (l  +  --^  cot  a  ) . 


ANSWERS  TO   THE   EXAMPLES  325 

CHAPTER  IX.,  p.  83. 

1.     196,384  miles  per  sec.  2.     1  mile  1226  yds.  2  ft.  3.     88. 

4.  (1)  17^;  20T5r.     (2)  18}.     (3)48.     (4)1:507.'  5.     11 -18  sec. 

6.  324  ft.  7.     256  ft. 

CHAPTER  X.,  p.  87. 
1.     1  :  289.  2.     25  :  9.  3.     12 -9  ft.  sec.  units. 

CHAPTER  XII.,  p.  114. 

1.     22;  29£;  52|;  5T\ ;  7£;  27-ft.  2.     210670. 

3.  18-391  miles  per  sec.  4.     1037*5  miles  an  hour. 

5.  27,777-7  :  911'4.  6.     774'4  yds. ;  25'2  mis.  an  hr;  62f  sec. 

7.  1717-3  yds.  8.     1098  ft.  per  sec.  9.     }}  ft.  sec.  units. 
10.     7-5.                11.     112;  208;  400.                12.     768. 

13.     After  3|  sec. ;  24  up  wards;  8  down;  104  down.  14.     1'79. 

15.     1-955;  220ft.  16.     (a)  320,000;  (6)  270,000. 

17.     1|  sec. ;  6 ;  510 ;  at  the  end  of  the  ten  seconds.  18.     u  + 1  (2w  -  1 ). 

19.     355  days  5  hrs;  2854170  million  miles.  20.     100  sec. ;  40,000  ft. 

21.     8163  metres.  22.     -2;   -2;  5  sec.;  12^  feet. 

CHAPTER  XIII.,  p.  134. 

3.     (1)  (a)  6;  (b)  4;  (2)  (a)  125;  (6)  222'46.      4.    24;  9;  1344;  };  £;  42. 
5.     16326-5.  6.     96  Ibs. ;  2|  Ibs. ;  1  Ib. ;  50  gms. 

7.    6673-8;  33369  cms.        8.    -24;  (a)  7'2;  (6)  35 -6;  10800  ft. ;  4  m.  56'6  sec. 
9.     3-094.  10.     75-3 ;  400  ft. ;  27ft  sec.  11.     4T^  tons ;  l^  tons. 

12.     625  Ibs.          13.     44-64  ft.  per  sec.         14.     1350000;  1318-3  Ibs. ;  3'6  in. 

15.  -12;  -1.  16.     9-76  Ibs.  17.     17  times.  18.     358  Ibs. 
19.    About  7°  17'.                20.     43'86  tons. 

CHAPTER  XIV.,  p.  141. 

1.     57-2;  457-8.  2.     980'4.  3.     #=15  gms. ;  #=978'9. 

4.  5£ ;  10|  ft.  5.     224-91  gms. ;  223  gms. ;  1§  Ibs. 

% 

CHAPTER  XV.,  p.  153. 

1.     (a)  82665779;  (6)81312000.  2.     89:1.  4.     ^5  H-p- 

5.  4£.  6.     29Jf.  7.     -10625.  8.     134-4. 

9.     1469  yds. ;  34f  miles.  10.     24'42  Ibs.  11.     18|  Ibs. 

12.     57600  Ibs.          13.     -00014178....          14.     107 '4.          15.     450,000  Ibs. 

16.  About  4,940,000.  17.     168-19  Ibs. 

21—3 


326  MECHANICS 

CHAPTER  XVI.,  p.  158. 
1.     61°  24'.  2.     8-8  in.  3.     182180  miles  per  sec. 

CHAPTER  XVII.,  p.  169. 

3.     60°  from  smaller  force.  8.     (1)  25  Ibs. ;  16°  16'  from  7  Ib.  force. 

(2)    13   Ibs.;     27°  48'   from   8   Ibs.       (3)    1276   Ibs.;     48°  15'   from    14   Ibs. 
(4)  12-61  Ibs. ;  22°  1'  from  8  Ibs.  9.     18 '00  Ibs. ;  70°  40'  from  7  Ibs. 

10.  9*66  Ibs. ;  96°  8'  36"  from  force  1  on  opposite  side  to  force  2. 

11.  45-96  Ibs.  12.     103-92  Ibs.  13.     500  Ibs. ;  300  Ibs. 

14.     20f  Ibs. ;  541  Ibs.  15.     9  Ibs. ;  12  Ibs.  19.     36-6  Ibs. ;  25 -8  Ibs. 

P  PO 

20.     4-62  Ibs. ;  8  Ibs.  24.     tan  6  =  -  ;     T  =  ~= 


Q  */P*  +  Q* 

25.     37  \  Ibs.  26.     62°  42';  11  Ibs. 


CHAPTER  XVIII.,  p.  189. 

3.     V^~  M^2.  4.     20-8  Ibs.  ;  70°  54'  to  horizon;  158  Ibs. 

6.     70°  53'  with  horizon  ;  26  -45  Ibs. 

acotct-6cot/3      TI7      sin/3  ,ir      sin  a 

8-  *"'-    -^+F~-;  W^^+W)'  *&&+$>• 

9.     Let  W  be  the  weight  of  the  rod  ;  6  its  inclination  to  the  vertical.     Then 

W  a-b 

T=-  —   —  ;         cot0  =  -  T  cot  a. 
2  cos  a'  a  +  b 


/y-(a 
V          ab 


where  cosa=    g     v          ^ 

10.      JFtana;  TFseca.  12.     30°  to  vertical;  11-55  Ibs. ;  577  Ibs. 

15.     28-3  Ibs.  16.     91  Ibs.  17.     102|  Ibs. ;  0.  18.     20  Ibs. 

20.     4 \/2.  21.     6  Ibs.  along  DA  produced ;  18-186  AB. 


CHAPTER  XIX.,  p.  203. 

1.     16-8  Ibs. ;  18-277  Ibs:                2.     -25.  3.     6  tons. 

4.     -115;  30°.            5.     791:1000.            6.     -577.  8.     About  12 J  tons 

CHAPTER  XX.,  p.  211. 

1.     200  ft. ;  10  sec.  2.     12£  sec. ;  34  miles  an  hour. 

3.     About  16  sec. ;  26  miles  an  hour ;  450  feet.  4.     8  ft.  per  sec. 

7.     2-23  sec.                8.     30°.                9.     6  oz.  10.     37  ft.  6  in. 


ANSWERS  TO   THE   EXAMPLES  327 

CHAPTER  XXI.,  p.  222. 

1.     2-5  sec.  ;  173-2  ft.  ;  25  ft.          2.     29'7  miles.          3.     2000  ft.  per  sec. 

4.  38°  19'  or  85°  26'.  5.     20-057  miles.  6.     109-75  ft.  per  sec. 

7.  176-7.  8.     86-4  ft.  per  sec.  ;  170-64  ft.  per  sec. 

10.     16-4  ft.  per  sec.  ;  31  ft.  Of  in.  11.     72-2  sec.;  15-8  miles. 

13.     37-02  ft.  per  sec. 

CHAPTER  XXIL,  p.  235. 

1.     (1)  1-57  sec.  ;  (2)  1-256  sec.  ;  (3)  1'57  sec.  2.     (1)  4  ft.  per  sec.  ; 

(2)  50  cm.  per  sec.  ;  (3)  8  ft.  per  sec.  4.     2'09  sec.  ;  117  ft.  /sec2. 

5.  27.  6.     -4014  sec.  ;  156-53  cm./sec.  7.     1,293,600  dynes. 

8.  2-5  sec.  ;  3*02  sec. 

CHAPTER  XXIII.,  p.  241. 

1.     9-78496  in.  2.     34".  3.     1/160  in.  4.     15-2  cm. 

5.  32-197.  6.     5-13.  7.     10-8.          8.     3666-6  ft.          9.     1098-8  ft. 

CHAPTER  XXIV.,  p.  255. 
5.     4332-5  days;  about  480  million  miles. 

CHAPTER  XXV.,  p.  267. 

1.     (1)  4-/  ft.  per  sec.;  5f  units.     (2)  ^=4^;  v2  =  5f  ;  9}  units. 
2.     vl=  -4;  v2=5£;  93|  foot-poundals.        5.     1^  sec.  ;  '88  inch  per  second. 

6.  331  Ibs.  7.     51400  Ibs.  8.     3  blows. 

10.  60°  57'  18"  ;  29°  2'  42".     The  directions  are  at  right  angles. 

11.  44*6  ft.  per  sec.  12.     The  axis  is  diminished  by  six-millionths, 
i.e.  about  550  miles.     The  year  is  shortened  by  4  min.  24  sec. 

CHAPTER  XXVI.,  p.  275. 
1.     6-26  inches.  2.     58-71  inches. 

CHAPTER  XXVIII.,  p.  287. 

1.     274.  2.     17-4  tons.  3.     348-52;  31  sees. 

4.     (1)  1-69  ft./sec2;  (2)  1788  ft./sec2.  5.     64-15. 


6.     (Complete  oscillation)  2ir  \  ;  10-37  inches.  7.     116-25. 

CHAPTER  XXIX.,  p.  302. 
1.     243-515.  2.     789-4  ft.  per  sec. 

CHAPTER  XXX.,  p.  312. 
1.     32-1906. 


INDEX. 


The  numbers  refer  to  the  pages. 


Acceleration,  80 

Of  falling  body,  81 

Definition,  111 

Unit,  111 

Geometrical  Representation,   111 

In  Circle,  226 
Adams,  95,  121,  122 
Airy,  315 

Angular  Momentum,  284,  285 
Angular  Motion,  283,  284 

Compared  with  linear,   284,  285 
Archimedes,  3 

Principle  of  Lever,  4,  66 
Attraction  of  a  Sphere 

Internal  point,  253 

External  point,  253-255 

Within  the  Earth,  255 
Atwood's  Machine,  136-141,  153 

Balance,  32 

Conditions  of  Sensitiveness,  33 

Description  of,  33 

Use  of,  35 

Method  of  Oscillations,  36 

Construction,  37 

Precautions  with,  37 

Borda's  method,  38 

Gauss'  method,  38 

Steelyard,  39 

Eoberval's,  61 
Balancing 

Bicycle,  29 

On  tight-rope,  29 
Ball,  Sir  Robert,  77,  197 
Ballistic  Galvanometer,  297-301 
Ballistic  Pendulum,  297,  298 
Benvenuto  Cellini,  238 
Bernouilli,  Principle  of  Virtual  Work, 

55 

Borda,  38 
Bouguer,  315 
Brachistochrones,  208-210 
Bradley,  120 

Capstan,  19,  179 
Carlini,  315 


Cavendish  Experiment,  312-323 

Result,  321,  322 

Other  results,  322 
Central  Forces,  243-255 

Equable  description  of  areas,  243-245 

Law  of  Force,  246 

Ellipse  about  Centre,  246-248 

Ellipse  about  Focus,  248-252 
Centre  of  Gravity,  23 

Definition,  24 

Experimental  determination,  25 

Rectangle,  26 

Straight  Line,  26 

Triangle,  26 

General  Formula  for,  27 
Centre  of  Oscillation,  304 
Centre  of  Percussion,  305,  306 
Centrifugal  force,  85, 
Clairaut,  95 

Coefficient  of  Restitution,  261-263 
Coincidences,  Method  of,  307-309 
Columbus,  120 
Component  159,  161,  162 
Compound  Pendulum,  271,  272 

Huyghens'  Solution,  272,  273 

By  D'Alembert's  Principle,  278-280 
Conditions  of  Equilibrium,  159 

Two  Forces,  165 

Three  Forces,  166-167 

Any  number,  167 

Coplanar,  173-187 
Conservation  of  Energy,  151 

Maxwell's  Statement,  152 
Constant  of  Gravitation,  313 
Co-ordinates 

Cartesian,  108 

Polar,  109 
Copernicus,  89,  120 
Coulomb,  194 
Couples,  179-183 

Work  done  by,  284,  285 
Cricket  Bat,  Theory  of,  305,  306 

D'Alembert,  95 

D'Alembert's  Principle,  276,  277 
Compound  Pendulum,  278,  279 


INDEX 


Density  of  the  Earth,  314,  315 

Newton's  Guess,  323 
Des  Cartes,  143,  144 

Vortices,  89 
Descent  of  Chord,  208 
Differential  Pulley,  60,  198 
Dynamics,  Definition  of,  2,  104 

Efficiency  of  a  Machine,  198 
Energy,  142-153 

Kinetic,  143 

Compared    with     Momentum,    143- 
145 

Potential,  145-146 

Transfer  of,  147 

Storage,  148 

Transformation,  148 

Regulation  of  Supply,  150 

Conservation  of,  151 

Sources,  152 

Energy  of  Eotation,  284,  285 
Equilibrium,  28 

Bicycle,  29 

Neutral,  29 

Stable,  29 

Tight-rope  dancer,  29 

Unstable,  29 

By  Virtual  Work,  153 

Conditions  of,  159 

Coplanar  Forces,  186-187 
Erg,  57 
Euler,  281 
Explanation,  nature  of,  8,  97,  98 

First  Law  of  Motion 

Discovery,  78 

Statement,  120 

Evidence  for,  121-122 

Verification,  138 
Flamsteed,  95 
Flywheel,  289-291 
Foot-pound,  57 
Foot-poundal,  57 
Force 

Definition,  12 

Idea  generalised,  98 

Denned,  98,  117 

Units,  124 

Poundal,  125 

Dyne,  126 
Forces 

Representation  of,  155 

Two  forces  at  a  point,  160 

Three  forces,  163 

Any  number,  164-165 

Triangle  of,  167 

Polygon  of,  167,  168 

Coplanar,  173-187 

Transmissibility,  174 

Parallel,  176 


Forces  (cont.) 

Eeduction  to  Couple  and  Force,  184- 

185 

Fresnel,  121 
Friction 

Determination,  193,  195 

Angle  of,  194 

Laws,  194 

Coefficient,  194 

Cone  of  195 

In  Simple  Machines,  197 

Hope  on  Post,  200 
Funicular  Machine,  49 

Galileo 

Inclined  Plane,  53 
Inclined  Plane  deduced  from  lever,  66 
Work  and  Equilibrium,  66,  273 
Problem  of  Falling  Bodies,  69 
Formula  of  Falling  Bodies,  72 
Experiments  on  Falling  Bodies,  73 
Velocity  due  to  vertical  fall,  75 
Pendulum  experiment,  76 
Velocity  acquired  per  second,  77 
First  Law  of  Motion,  78,  89 
The  Pendulum,  82,  238 
Tower  of  Pisa,  126 

Gauss,  38 

Girard,  Albert,  69 

Halley,  96,  97 

Hamilton,  Sir  W.  B.,  121 

Hammer,  149 

Helmholtz,  151 

Hipparchus,  95 

Hodograph,  225,  226 

Hooke,  96,  97,  250 

Horse  and  Cart,  Problem  of,  129-132 

Horse  Power,  152 

Huyghens 

Motion  in  a  Circle,  84 
Acceleration  in  a  Circle,  85 
Formulae  for  a  Circle,  86 
Horologium  Oscillatorium,  88,  276 
Pendulum  Clock,  238 
Compound  Pendulum,  272-275 
Centre  of  Oscillation,  304 
Centre  of  Percussion,  305,  306 

Impact,  257-266 

Inelastic  Bodies,  258,  259 

Elastic  Bodies,  259-261 

Loss  of  Energy  in,  263,  264 

Of  Stream,  265 

On  Fixed  Plane,  265 

Oblique,  266 
Impulse,  Definition,  123 
Impulsive  Forces,  257-266 
Inclined  Plane 

Principle  of,  41,  66 


330 


INDEX 


Inclined  plane  (cont.) 

Stevinus'  proof,  42 

Experimental  Verification,  44 

Criticism  of  proof,  45 

Mach,  46 

Deduced  from  Work,  53 

Motion  on,  207,  208 
Inertia,  99 
Isochronous  Vibrations,  234 

Joule,  151 

Eater's  Pendulum,  303-313 

Construction,  306,  307 

Length,  307 

Time  of  Swing,  307-309 
Kepler 

Laws,  89 

Second  Law,  245 

First  Law,  250 

Third  Law,  251 
Kilogramme,  118 

In  pounds,  118 
Kinematic  Formulae 

Uniform  motion,  111 

Uniform  Acceleration,  112,  113,  125 
Kinematics,  Definition  of,  104 
Kinetic  Theory  of  Gases,  265 
Kinetics,  Definition  of,  104 
Knife,  Theory  of,  199 

Lagrange,  Proof  of  Principle  of  Virtual 

Work,  63 

Lami's  Theorem,  167,  186 
Laws  of  Motion,  116-134 

Experimental  Verification,  136 
Laws  of  Nature,  Establishment  of,  120 
Leonardo  da  Vinci,  Moment  of  a  Force, 

13,  178,  286 
Lever,  Principle  of,  3,  178 

Archimedes'  proof,  4 

Stevinus'  proof,  6 

Galileo,  6,  66 

Lagrange,  6 

Criticism  of  proof,  7 

Experimental  proof,  10 

Classes  of,  15 

The  forearm,  17 

The  oar,  17 

The  shadoof,  17 

Deduced  from  Lever,  66 

Deduced  from  Parallelogram  of  Forces, 
67 

Deduced  from  Work,  68 
Leverrier,  121,  122 
Logarithmic  Law,  202 
Lunar  Theory,  95 

Mach,  3 

Principle  of  Lever,  7 


Mach  (cont.) 

Moment  of  Force,  13 

Inclined  Plane,  46 

Instinctive  knowledge,  46 

Galileo  and  the  Inclined  Plane,  54 

Galileo  and  Falling  Bodies,  70 

First  Law  of  Motion,  78 
Magnetometer  297 
Maskelyne,  315 
Mass 

Concept  of,  98 

Distinguished  from  Weight,  99,  119 

Definition,  117 

Test  of  Equality,  118 

Units,  118 

Maupertuis,  Criterion  of  Equilibrium,  65 
Maxwell,  105,  107,  151 
Maxwell's  Needle,  294-296 
Mayer,  151 
Measurement,  105 
Mechanical  Advantage,  15 

Of  Lever,  15 

Mechanics,  Meaning  and  Origin  of,  1 
Mendenhall,  316 

Method  of  Coincidences,  307-309 
Metre,  106 

In  inches,  106 

In  wave-lengths,  107 
Michell,  Kev.  John,  316 
Michelson,  107 
Micron,  106 

Modulus  of  Torsion,  294-296 
Moment  of  a  Force,  13,  286 

Definition  of,  14 

Geometrical  Representation,  176 
Moment  of  Momentum,  285 
Moments  of  Inertia,  281-293 

Definition,  281 

Table  of,  281,  282 
Momentum,  Definition,  123 
Morin's  Machine,  213-216 
Motion 

Definition,  110 

Eelativity  of,  110 
Motion  on  Inclined  Plane,  207,  208 

Down  Curve,  210 

In  Vertical  Circle,  210,  211 

Nasmyth,  264,  265 
Nautical  Almanac,  122,  127 
Neptune,  121,  122 
Newton 

Parallelogram  of  Forces,  50,  128 

Birth  and  Education,  90 

Career  and  discoveries,  90 

Law  of  Gravitation,  91 

Verified  for  Moon,  93 

Writing  the  Principia,  94 

The  Lunar  inequalities,  95 

Tides,  96 


INDEX 


331 


Newton  (cont.) 

Principia  published,  97 
Laws  of  Motion,  116-134 
Definition  of  Force,  117 
Guinea  and  Feather,  127 
Central  Forces,  243 
Attractions  of  Spheres,  252-255 
Laws  of  Impact,  259-261 
Density  of  the  Earth,  323 

Oerstedt,  7 

Parabolic  Motion,  213-222 
Parallel  Forces,  176,  177 
Parallelogram  of  Forces,  98,  128,  158 

Stevinus,  48 

Experimental  Verification,  50 

Newton,  50 
Parallelogram  Law 

Displacements,  156 

Velocities,  156 

Accelerations,  158 

Forces,  158 

Picard's  value  of  a  degree,  93 
Pile  driving,  264,  265 
Poinsot,  179 
Polygon  of  Forces,  167 
Pound,  Imperial,  118 

In  Kilogrammes,  118 
Principia,  The 

Writing,  94 

Contents,  95,  96 

Published,  97 
Problems,  Method  of  Solving 

Dynamical,  124 

Statical,  187 
Projectile,  Path  of,  81 
Projectiles,  213-222 

Time  of  Flight,  219 

Eange,  219,  220 

Greatest  Height,  220 

Kange  on  Slope,  220-222 
Ptolemy,  95 
Pulley,  19 

Archimedes'  System,  20 

Pulley  Block,  21 
Punching  Machine,  149 

Kelative  Motion,  157 
Resolved  Part,  162 
Resultant,  158 
Reversible  Processes,  151 
Roberval's  Balance,  61 
Royal  Society,  96,  97,  303,  316 

Scalar  quantities,  155 
Second  Law  of  Motion 

Statement,  122 

Formula,  123,  125 

Independence  of  Forces,  127 


Second  Law  of  Motion  (cont.) 
Parallelogram  of  Forces,  128 
Verification,   138 

Simple  Equivalent  Pendulum,  272 

Simple  Harmonic  Motion,  224-234 
Definition,  224 
Acceleration  in,  227 
Law  of  Force,  227,  228 
Experiments  on,  228,  229 
Amplitude,  229 
Period,  229 
Phase,  229 
Epoch,  230 

Composition  of,  230-232 
Resolution,  232-234 
Fundamental  Property,  234 

Simple  Pendulum,  237-241 
Isochronous,  238,  239 
Seconds  Pendulum,  239,  303 
Effect  of  change  of  gravity,  239 
Effect  of  change  of  length,  239 
Measurement  of  Depths,  240 
Measurement  of  Heights,  240 
Value  of  Gravity,  240,  241 

Standards,  International,  105 
Length,  106,  107 
Time,  107 
Mass,  118 

Statics,  Definition  of,  2,  104 

Steam  Hammer,  264,  265 

Steelyard,  39 

Stevinus 

Principle  of  Lever,  6 
Inclined  Plane,  41,  66 
Parallelogram  of  Forces,  48 
Principle  of  Virtual  Work,  52 

Sufficient  Reason,  Principle  of,  7 

Testing  Engine 

Buckton,  16 

Emery,   18 
Third  Law  of  Motion,  99,  122 

Verification,  140 

Torque,  14,  178,  179,  181,  182,  286 
Torricelli,   Principle   of  Virtual  Work, 

54 

Torsion  Pendulum,  291-293,  317,  318 
Transmissibility  of  Force,  174 
Triangle  of  Forces,  167,  186 
Tycho  Brahe,  95 

Units 

Length,  106 
Time,  107 
Velocity,  110 
Acceleration,  111 
Mass,  118 
Momentum,  123 
Impulse,  123 
Force,  124,  125,  126 


332 


INDEX 


Uranus,  121 

Value  of  Gravity,  310-312 
Varignon's  Theorem,  175 
Vectors,  156 
Velocity 

Due  to  vertical  fall,  75 

Acquired  per  second  of  fall,  77 

Idea  of  variable,  79 

Definition,  110 

Unit,  110 

Variable,  110 

Geometrical  Representation,  111 

Of  bullet,  297,  298 
Virtual  Work 

Stevinus,  52 

Principle  of,  52,  153 

Galileo,  53 

Mach,  54 

Torricelli,  54 

Bernouilli,  55 

Statement  of  Principle,  58 

Differential  Pulley,  60 

Screw,  60 


Virtual  Work  (cont.) 
Eoberval's  Balance.  61 
Lazy  Tongs,  62 
Lagrange's  Proof,  63 
Criterion  of  Equilibrium,  65 
Deduced  from  Lever,  68 

Watt,  153 
Watt,  James,  152 
Wedge,  198 
Weight 

Distinguished  from  Mass,  119 

Proportional  to  Mass,  126-127 
Wheel  and  Axle,  19 
Work,  142-153 

Nature  of,  55 

Factors  of,  56 

Definition,  57 

Units  of,  57 

Galileo,  66 
Wren,  96,  97 

Yard,  Imperial,  106 
Young,  Thomas,  121 


CAMBRIDGE  :    PRINTED    BY   JOHN    CLAY,    M.A.    AT    THE    UNIVERSITY    PRESS. 


CAMBRIDGE    PHYSICAL 
SERIES 


The  Times. — "The  Cambridge  Physical  Series... has  the  merit  of  being 
written  by  scholars  who  have  taken  the  trouble  to  acquaint  themselves  with 
modern  needs  as  well  as  with  modern  theories." 

Conduction  of  Electricity  through  Gases.     By  Sir  J.  J. 

THOMSON,  D.Sc.,  LL.D.,  Ph.D.,  F.R.S.,  Fellow  of  Trinity  College 
and  Cavendish  Professor  of  Experimental  Physics,  Cambridge,  Pro- 
fessor of  Natural  Philosophy  at  the  Royal  Institution,  London.  Second 
Edition  enlarged  and  partly  re- written.  Demy  8vo.  viii  +  678  pp.  i6s. 


CONTENTS. 


I. 


II. 
III. 


IV. 


V. 


VI. 


VII. 
VIII. 


IX. 


Conductivity    of 
in      a       normal 


Electrical 

Gases 

state. 
Properties  of  a  Gas  when  in 

the  conducting  state. 
Mathematical  Theory  of  the 

Conduction  of  Electricity 

through  a  Gas  containing 

Ions. 

Effect  produced  by  a  Mag- 
netic Field  on  the  Motion 

of  the  Ions. 
Determination  of  the  -Ratio 

of  the  Charge  to  the  Mass 

of  an  Ion. 
Determination  of  the  Charge 

carried   by  the  Negative 

Ion. 
On  some  Physical  Properties 

of  Gaseous  Ions, 
lonisation  by  Incandescent 

Solids, 
lonisation    in    Gases    from 


X. 

XI. 
XII. 
XIII. 


XIV. 


XV. 
XVI. 

XVII. 

XVIII. 
XIX. 
XX. 
XXI. 


lonisation  by  Light. 
Photo- Electric  Effects. 

lonisation  by  Rontgen 
Rays. 

Rays  from  Radio-active 
Substances. 

Power  of  the  Elements  in 
general  to  emit  ionising 
radiation. 

lonisation  due  to  Chemical 
Action,  the  bubbling  of 
air  through  water,  and 
the  splashing  of  drops. 

Spark  Discharge. 

Discharge  through  Gases 
at  Low  Pressures. 

Theory  of  the  Discharge 
through  Vacuum  Tubes. 

The  Electric  Arc. 

Cathode  Rays. 

Rontgen  Rays. 

Properties  of  Moving  Elec- 
trified Bodies. 

Index. 


Flames. 

"It  is  difficult  to  think  of  a  single  branch  of  the  physical  sciences  in 
which  these  advances  are  not  of  fundamental  importance.  The  physicist 
sees  the  relations  between  electricity  and  matter  laid  bare  in  a  manner 
hardly  hoped  for  hitherto. ...The  workers  in  the  field  of  science  are  to-day 
reaping  an  unparalleled  harvest,  and  we  may  congratulate  ourselves  that 
in  this  field  at  least  we  more  than  hold  our  own  among  the  nations  of 
the  world."  Times  (on  the  First  Edition) 

A  Treatise  on  the  Theory  of  Solution,  including  the 

Phenomena  of  Electrolysis.  By  WILLIAM  CECIL  DAMPIER  WHETHAM, 
M.A.,  F.R.S.,  Fellow  of  Trinity  College.  Demy  8vo.  x  +  488  pp. 
IQS.  net. 

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"The  treatment  throughout  is  characterised  by  great  clearness,  especially 
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CAMBRIDGE  PHYSICAL  SERIES 


Mechanics  and  Hydrostatics.  An  Elementary  Text- 
book, Theoretical  and  Practical,  for  Colleges  and  Schools.  By 
R.  T.  GLAZEBROOK,  M.A.,  F.R.S.,  Director  of  the  National  Physical 
Laboratory  and  Fellow  of  Trinity  College,  Cambridge.  Crown  8vo. 
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Also  in  separate  volumes 

Part  I.     Dynamics.     256  pp.     %s. 

Part  II.     Statics.     182  pp.     2s. 

Part  III.     Hydrostatics.     216  pp.     2J. 

Extract  from  Preface-  It  has  now  come  to  be  generally  recognized 
that  the  most  satisfactory  method  of  teaching  the  Natural  'Sciences  is  by 
experiments  which  can  be  performed  by  the  learners  themselves.  In 
consequence  many  teachers  have  arranged  for  their  pupils  courses  of 
practical  instruction  designed  to  illustrate  the  fundamental  principles  of 
the  subject  they  teach.  The  portions  of  the  following  book  designated 
EXPERIMENTS  have  for  the  most  part  been  in  use  for  some  time  as  a 
Practical  Course  for  Medical  Students  at  the  Cavendish  Laboratory. 

The  rest  of  the  book  contains  the  explanation  of  the  theory  of  those 
experiments,  and  an  account  of  the  deductions  from  them.  This  part  has 
grown  out  of  my  lectures  to  the  same  class.  It  has  been  my  object  in  the 
lectures  to  avoid  elaborate  apparatus  and  to  make  the  whole  as  simple  as 
possible.  Most  of  the  lecture  experiments  are  performed  with  the  apparatus 
which  is  afterwards  used  by  the  class,  and  whenever  it  can  be  done  the 
theoretical  consequences  are  deduced  from  the  results  of  these  experiments. 

It  is  with  the  hope  of  extending  some  such  system  as  this  in  colleges 
and  schools  that  I  have  undertaken  the  publication  of  the  present  book  and 
others  of  the  series. 

Extracts  from  Press  Notices 

"  Schools  and  Colleges  will  certainly  benefit  by  adopting  this  book  for 
their  students."  Nature 

"Mr  Glazebrook's  volumes  on  Heat  and  Light  deal  with  these  subjects 
from  the  experimental  side  and  it  is  difficult  to  admire  sufficiently  the 
ingenuity  and  simplicity  of  many  of  the  experiments  without  losing  sight 
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Review 

"The  books  almost  cover  the  advanced  stages  of  the  South  Kensing- 
ton prospectus  and  their  use  can  certainly  be  recommended  to  all  who  wish 
to  study  these  subjects  with  intelligence  and  thoroughness."  Schoolmaster 

"The  book  is  very  simply  and  concisely  written,  is  clear  and  methodic 
in  arrangement.... We  recommend  the  book  to  the  attention  of  all 
students  and  teachers  of  this  branch  of  physical  science."  Educational 
News 

' '  It  will  be  especially  appreciated  by  teachers  who  possess  the  necessary 
apparatus  for  experimental  illustrations."  Athenaum 

"Text-books  on  this  subject  are  generally  too  simple  or  too  elaborate 
for  a  conception  of  elementary  mechanical  principles.  This  book  cannot 
fail  to  recommend  itself  therefore  for  a  first  course  preliminary  to  the  study 
of  physical  science.  No  other  book  presents  in  the  same  space  with  the 
same  clearness  and  exactness  so  large  a  range  of  mechanical  principles." 
Physical  Review 

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Statics  in  the  present  volume — The  collected  examples  for  students' 
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CAMBRIDGE  PHYSICAL  SERIES 


Heat  and  Light.  An  Elementary  Text-book,  Theoretical 
and  Practical,  for  Colleges  and  Schools.  By  R.  T.  GLAZEBROOK, 
M.A.,  F.R.S.  Crown  8vo.  5*. 

Also  in  separate  volumes  : 

Heat.     230  pp.     3-r. 
Light.     213  pp.     3.5-. 

Electricity  and  Magnetism :  an  Elementary  Text-book, 
Theoretical  and  Practical.  By  R.  T.  GLAZEBROOK,  M.A.,  F.R.S., 
Crown  8vo.  Cloth,  i — 440  pp.  6s. 

Extract  from  Preface.  Some  words  are  perhaps  necessary  to  explain 
the  publication  of  another  book  dealing  with  Elementary  Electricity. 
A  considerable  portion  of  the  present  work  has  been  in  type  for  a  long 
time ;  it  was  used  originally  as  a  part  of  the  practical  work  in  Physics  for 
Medical  Students  at  the  Cavendish  Laboratory  in  connexion  with  my 
lectures,  and  was  expanded  by  Mr  Wilberforce  and  Mr  Fitzpatrick  in  one 
of  their  Laboratory  Note-books  of  Practical  Physics. 

When  I  ceased  to  deliver  the  first  year  course  I  was  asked  to  print  my 
lectures  for  the  use,  primarily,  of  the  Students  attending  the  practical 
classes  ;  the  lectures  on  Mechanics,  Heat  and  Light  have  been  in  type  for 
some  years.  Other  claims  on  my  time  have  prevented  the  issue  of  the 
present  volume  until  now,  when  it  appears  in  response  to  the  promise  made 
several  years  ago. 

Meanwhile  the  subject  has  changed  ;  but  while  this  is  the  case  the 
elementary  laws  and  measurements  on  which  the  science  is  based  remain 
unaltered,  and  I  trust  the  book  may  be  found  of  service  to  others  besides 
my  successors  at  the  Cavendish  Laboratory. 

The  book  is  to  be  used  in  the  same  way  as  its  predecessors.  The  appa- 
ratus for  most  of  the  Experiments  is  of  a  simple  character  and  can  be 
supplied  at  no  great  expense  in  considerable  quantities. 

Thus  the  Experiments  should  all,  as  far  as  possible,  be  carried  out  by 
the  members  of  the  class,  the  teacher  should  base  his  reasoning  on  the 
results  actually  obtained  by  his  pupils.  Ten  or  twelve  years  ago  this 
method  was  far  from  common  ;  the  importance  to  a  School  of  a  Physical 
Laboratory  is  now  more  generally  recognized  ;  it  is  with  the  hope  that  the 
book  may  be  of  value  to  those  who  are  endeavouring  to  put  the  method  in 
practice  that  it  is  issued  now. 

Extracts  from  Press  Notices 

"  If  the  nature  of  the  book  be  taken  into  consideration,  it  will  be  found 
unusually  free  from  the  influence  of  the  examination  spirit.  The  writing 
is  bright  and  interesting,  and  will  stimulate  a  desire,  we  think,  for  further 
study."  Athenceum 

'*•  Every  schoolmaster  and  teacher  who  has  under  consideration  the 
selection  of  a  text-book  for  his  better  students  should  most  certainly 
look  into  this  book.  The  information  is  everywhere  absolutely  sound 
and  reliable."  Guardian 

"  In  this  volume  we  have  a  thorough  analysis  of  the  elementary  laws 
of  electricity  and  magnetism ;  but  coupled  with  it  there  are  excellent 
expositions  of  the  operation  of  these  laws  in  practice.  The  book,  in  fact, 
is  in  many  respects  a  manual  for  the  laboratory  as  well  as  a  work  for 
the  class-room.... It  is  the  work  of  one  who  has  had  long  experience  as  a 
teacher  in  an  important  centre  of  education,  and  all  who  study  its  contents 
carefully  and  intelligently  will  acquire  a  sound  knowledge  of  the  principles 
of  the  attractive  subject  with  which  it  deals."  Engineering 


CAMBRIDGE  PHYSICAL 
SERIES 


General  Editors :  F.  H.  NEVILLE,  M.A.,  F.R.S.  and 

W.  C.  D.  WHETHAM,  M.A.,  F.R.S. 
Modern  Electrical  Theory.     By  N.  R.  CAMPBELL,  M.A. 

Demy  8vo.     pp.  xii +  332.     7.?.  6d.  net. 

Mechanics.  By  JOHN  Cox,  M.A.,  F.R.S.C.,  Macdonald 
Professor  of  Experimental  Physics  in  McGill  University,  Montreal. 
Demy  8vo.  pp.  xiv  +  332.  With  four  plates  and  numerous  figures. 
9-r.  net. 

The  Study  of  Chemical  Composition.     An  Account  of 

its  Method  and  Historical  Development,  with  illustrative  quotations. 

By  IDA  FREUND,  Staff"  Lecturer  and  Associate  of  Newnham  College. 

Demy  8vo.     xvi  +  6so  pp.     i8j.  net. 

"  Written  from  a  broad,  philosophical  standpoint,  we  know  of  no  book 
more  suited  for  the  student  of  chemistry  who  has  attained  a  sound  general 
knowledge  of  the  science,  and  is  now  ready  to  appreciate  a  critical  discus- 
sion of  the  methods  by  which  the  results  he  has  learnt  have  been  built  up, 
thereby  fitting  himself  for  the  real  world  of  investigation  on  his  own 
account."  Saturday  Review 

A  Treatise  on  the  Theory  of  Alternating  Currents. 

By  ALEXANDER  RUSSELL,  M.  A. ,M.  I. E.E.  In  two  volumes.  DemySvo. 

Vol.1,    pp.  xii  +  4o8.    i2J.net.     Vol.11,    pp.  xii  +  488.    i2j.net. 
"The  volume  is  not  only  rich  in  its  own  substantive  teaching,  but  well 
supplied  with  references  to  the  more  remote  authorities  upon  its  subject. 
It  opens  an  important  and  valuable  contribution  to  the  theoretical  litera- 
ture of  electrical  engineering."     Scotsman 

Radio-activity.  By  E.  RUTHERFORD,  D.Sc., F.R.S.,  F.R.S.C., 

Professor  of  Physics,  Victoria  University  of  Manchester.     Demy  8vo. 

Second  Edition.  Revised  and  enlarged.  xiv  +  .s8opp.  i2j.6of.net. 
"  Professor  Rutherford's  book  has  no  rival  as  an  authoritative  exposition 
of  what  is  known  of  the  properties  of  radio-active  bodies.  A  very  large 
share  of  that  knowledge  is  due  to  the  author  himself.  His  amazing 
activity  in  this  field  has  excited  universal  admiration."  Nature 
Experimental  Elasticity.  A  Manual  for  the  Laboratory. 

ByG.  F.  C.  SEARLE,  M.A.,  F.R.S.    DemySvo.    pp.xvi+i87.    5J.net. 
Air  Currents  and  the  Laws  of  Ventilation.    Lectures 

•  on  the  Physics  of  the  Ventilation  of  Buildings.  By  W.  N.  SHAW, 
Sc.D.,  F.R.S.,  Fellow  of  Emmanuel  College,  Director  of  the  Meteo- 
rological Office.  Demy  8vo.  pp.  xii  +  94.  3J.  net. 

The     Theory     of     Experimental     Electricity.       By 

W.  C.  D.  WHETHAM,  M. A.,  F.R.S.  DemySvo.  xii +  334  pp.  8j.net. 
"We  strongly  recommend  this  book  to  those  University  Students 
-who  require  an  introduction  to  modern  scientific  electricity.  The  research 
.atmosphere  of  Cambridge  is  here  brought  before  us,  and  the  student  is 
guided  into  the  paths  along  which  great  progress  has  been  made  in  recent 
years."  Guardian  

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